refractive phenomena in the shock wave dispersion with variable gradients

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Refractive phenomena in the shock wave dispersion with variable gradients A. Markhotok and S. Popovic Citation: Journal of Applied Physics 107, 123302 (2010); doi: 10.1063/1.3432565 View online: http://dx.doi.org/10.1063/1.3432565 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Traveling waves in Hall-magnetohydrodynamics and the ion-acoustic shock structure Phys. Plasmas 21, 022109 (2014); 10.1063/1.4862035 Shock wave refraction enhancing conditions on an extended interface Phys. Plasmas 20, 043506 (2013); 10.1063/1.4799541 Stability of shock waves in high temperature plasmas J. Appl. Phys. 110, 083512 (2011); 10.1063/1.3653253 Nonplanar electrostatic shock waves in dense plasmas Phys. Plasmas 17, 022303 (2010); 10.1063/1.3309733 Hydrodynamics of shock waves with reflected particles. I. Rankine-Hugoniot relations and stationary solutions Phys. Plasmas 13, 082112 (2006); 10.1063/1.2336185 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Thu, 18 Dec 2014 06:44:27

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Page 1: Refractive phenomena in the shock wave dispersion with variable gradients

Refractive phenomena in the shock wave dispersion with variable gradientsA. Markhotok and S. Popovic Citation: Journal of Applied Physics 107, 123302 (2010); doi: 10.1063/1.3432565 View online: http://dx.doi.org/10.1063/1.3432565 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Traveling waves in Hall-magnetohydrodynamics and the ion-acoustic shock structure Phys. Plasmas 21, 022109 (2014); 10.1063/1.4862035 Shock wave refraction enhancing conditions on an extended interface Phys. Plasmas 20, 043506 (2013); 10.1063/1.4799541 Stability of shock waves in high temperature plasmas J. Appl. Phys. 110, 083512 (2011); 10.1063/1.3653253 Nonplanar electrostatic shock waves in dense plasmas Phys. Plasmas 17, 022303 (2010); 10.1063/1.3309733 Hydrodynamics of shock waves with reflected particles. I. Rankine-Hugoniot relations and stationary solutions Phys. Plasmas 13, 082112 (2006); 10.1063/1.2336185

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Page 2: Refractive phenomena in the shock wave dispersion with variable gradients

Refractive phenomena in the shock wave dispersion with variablegradients

A. Markhotok and S. Popovica�

Department of Physics, Old Dominion University, Norfolk, Virginia, 23529, USA

�Received 9 March 2010; accepted 15 April 2010; published online 16 June 2010�

In this article the refraction effects in the weak shock wave �SW� dispersion on an interface with atemperature variation between two mediums are described. In the case of a finite-gradient boundary,the effect of the SW dispersion is remarkably stronger than in the case of a step change inparameters. In the former case the vertical component of velocity for the transmitted SW �therefraction effect� must be taken into account. Results of comparative calculations based on thetwo-dimensional model corrected for the refraction effect show significant differences in the shapesof the dispersed SW fronts. © 2010 American Institute of Physics. �doi:10.1063/1.3432565�

I. INTRODUCTION

Investigations of the shock wave �SW� interaction with aweakly ionized gas have been done intensively in the pastdecades and have shown promising results with respect topractical benefits for aircraft design and its impact on theenvironment. The results indicate broad possibilities tomodify and control the SW front.1–3 Any capability tomodify the structure of a SW can reduce its strength andsignificantly decrease the shock induced drag, thus mitigat-ing the effects of shocks in supersonic flight, heating prob-lems during re-entry into the atmosphere, and reducing theeffects of sonic booms. The changes in the SW front, such asits curvature, can possibly affect the structure of the detona-tion wave by affecting the ignition conditions through thedependence on the detonation speed.4,5

In Refs. 6–8 a new mechanism of the weak SW disper-sion related to the curved shape of the incident SW front or aboundary between two mediums of different temperatureshas been described. The need to bring up this mechanismwas to explain the odd shapes of the deflection signals fromthe SW as it entered the hot discharge regions. The advan-tage of the proposed model is that, to some extent, it offers arather simple method of calculations to predict the shape ofthe dispersed shock front. This cuts the computational costsubstantially by eliminating the need to employ laboriousand sophisticated numerical procedures. Because of the sim-plicity the results are free of computational ghosts or associ-ated numerical oscillations which may substantially contami-nate numerical solutions. The possibility that a shock frontincident on a hot plasma region may have a shape that is notnecessarily flat but rather has a definite curvature was con-sidered. The curvature of the incident shock front itself,and/or curvature of the boundary between neutral gas and thedischarge region, can lead to the change in the shock frontshape. This happens because when a shock front crosses aboundary its front and rear parts propagate in areas withdifferent temperatures and hence, different propagation ve-locities. As a result, the front part of the SW accelerates with

respect to the rear. Since, this leads to a modification of theshock front, the described mechanism opens a broad possi-bility to control weak SW dispersion by employing thissingle mechanism or, for example, a few different thermalmechanisms.8 In this article we consider the same effect spe-cifically on a nonsharp interface, i.e., when parameters of thetwo mediums change across the interface not abruptly butrather over some distance of at least several lengths of thefree mean pass of the gas molecules �finite gradient case�.

Such a situation appears more realistic for most applica-tions, and what is also important, the SW dispersion in thiscase becomes stronger. This reduction in strength occurs be-cause when a SW is incident on a sharp interface, some partof the incident shock energy gets reflected back. Thus, theenergy of the shock transmitted into the hot discharge areabecomes reduced compared to the incident energy. However,when the interface smoothly extends over the final distance,the reflected wave vanishes, and no SW energy loss occursduring its transmission through the interface.9 More strictly,the smoothness of the gradient can be defined based on theinvestigations.10 A plane shock moves into a region of gaswhich is initially at rest under uniform pressure but with anonuniform density was considered there. The applicabilitylimits for the gradient smoothness �the characteristic lengthof the extended boundary� can be determined from the ana-lyzing the solution for a constant shock strength. It wasfound that no solution exists for a density changing morerapidly than 1 /x2 where x is the longitudinal coordinate.

The experimental evidence of such a situation can befound in Ref. 11, where no shock reflections were observedoff the boundary with a weakly ionized plasma of a laserspark in air. Similarly, no reflected shock off the boundarywas seen when a SW was passed through a heated layer ofair around a hot tube. The boundary of the heated layer wasstill clearly observable.

Thus the Mach number of the transmitted shock M2 willbe the same as for the incident SW M1, i.e., their ratioM2 /M1 in the dispersion relation6–8 is equal to the unity. Inthis case the ratio of the shock velocities V2 and V1 in the hotand cold areas, respectively,a�Electronic mail: [email protected].

JOURNAL OF APPLIED PHYSICS 107, 123302 �2010�

0021-8979/2010/107�12�/123302/4/$30.00 © 2010 American Institute of Physics107, 123302-1

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Page 3: Refractive phenomena in the shock wave dispersion with variable gradients

V2/V1 = �T2/T1�M2/M1� ,

will be dependent on the ratio of the temperatures only. Sincein the case of a sharp interface the Mach numbers ratio isalways less than unity, for a smooth interface the dispersioneffect becomes significantly stronger and the shape of theweak shock front will undergo greater changes in the longi-tudinal direction,7 proportionally to this velocities ratio�V2 /V1�

Xi = �V2/V1 − 1��R − xi� ,

here R is the radius of the spherical incident shock front oran interface, and Xi and xi are coordinates of the transmittedand the incident shock fronts �or the interface�, Fig. 2.

Stronger dispersion in this case suggests paying moreattention to a secondary effect associated with the verticalvelocity component arising as the shock penetrates throughthe interface. This vertical component is responsible for thechange in the shock front in the transverse direction and isreferred to here as the refraction. In the next part of thisarticle we derive all relevant relations. Then, the model willbe applied to numerically calculate the SW dispersion on aplasma sphere. Comparing the results of calculations for thecases of sharp and finite-gradient boundaries demonstratesthe strength of the effect.

II. THE REFRACTION EFFECT

The refraction effect in the SW dispersion arises whenthere is a curvature difference between the incident shockfront and the interface7 between two mediums �1 and 2� witha temperature step T1 /T2. Thus, the refraction effect is aconsequence of the SW dispersion on the boundary. Considera planar shock front traveling from left to the right and inci-dent on a plasma sphere surrounded by a cold gas �Fig. 1�.This is a two-dimensional �2D� problem since each point atthe incoming flat shock front is incident on the plasma inter-face at different angles and only one point at the symmetryaxis is incident normally to the surface. Consider a point i onthe shock front that is incident on the plasma boundary underan angle � �between the incident velocity V1 and the normalto the interface�.

Since the tangential component of the total velocity ofthe shock front does not change through the boundary but thenormal component does, the total velocity of the shock frontchanges its direction by an angle � �Fig. 1�

� = � − tan−1�Vt/V2n� . �1�

This effect is stronger at larger incidence angles, so it willaffect more far off-axis points on the sphere with no effect onthe axis. If the temperature of the plasma region is higherthan in the surrounding gas, then the transmitted normal ve-locity component is higher than the incident, and the totalvector deflects toward the longitudinal symmetry axis of thesphere. Then the x-component of the shock front velocity inthe plasma region is responsible for the change in the front inthe longitudinal direction, and a small y-component induceschanges in the vertical �transverse� direction. Thisy-component of the transmitted velocity is the origin of thephenomena referred to here as refraction, i.e., the deflectionof the velocity vector from its initial, incident direction, inthe SW dispersion on a curved boundary. This effect is verysimilar to the optical refraction of a light wave on a boundarybetween two media in which the light propagation speedsdiffer.

Then, the derivation for the X-coordinate andY-coordinate of the dispersed shock front surface at the pointi can be done in the same way as it was done in Ref. 7. Onecan get a relationship between the coordinates of the trans-mitted shock front �Xi ,Yi� and the boundary surface �xi ,yi�,�Fig. 2�

Xi = xi�V1 − V2x�/V1 + t�V2x − V1�, Yi = yi − V2y�t −xi

V1� .

�2�

In this figure, the shock front �the most left line in bold�is coming from the left to the right and incident on thespherical plasma boundary �one quarter of the boundary isshown�. A part of the shock that propagates through theplasma gets curved �the second curved line inside the plasmasphere and crossing the plasma boundary� while the rest ofthe shock that continues traveling outside the plasma staysflat �the bold line outside of the boundary connected with thecurved part of the shock�.

Here time t starts at the first contact point between theincident shock front and the plasma boundary. Then, for ex-

V1n

V1t

V2n

V1

V2

γ

V1t

FIG. 1. Velocity components of the planar shock front incident on a plasmaand the transmitted shock front. The temperature of plasma T2 is higher thanthe temperature of the surrounding gas T1.

V1t

x i

y i

R

X i

ii

Yi

FIG. 2. Schematic diagram for the initially flat shock front change during itspenetration of the spherical interface. The incoming flat shock front �in bold�moves from the left to the right. The coordinates �xi ,yi� and �Xi ,Yi� are fora point i at the interface and the transmitted shock front correspondingly.

123302-2 A. Markhotok and S. Popovic J. Appl. Phys. 107, 123302 �2010�

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ample, for the time t=R /V1 needed to propagate the SW intothe cold area a distance equal to the radius R, the penetratedshock front coordinates are

Xi = �V2x

V1− 1��R − xi� = �V2

V1cos � − 1��R − xi� ,

Yi = yi −V2y

V1�R − xi� = yi −

V2

V1sin ��R − xi� . �3�

To keep further derivations more general, we first deriveexpressions for the case of a sharp boundary that can beextended for a more simple case of a nonsharp boundary.The ratio of velocities V2 /V1 in the system �2� can be easilyobtained by expressing the velocities components in terms ofthe incidence angle �, the ratios of the Mach numbers, andthe temperatures in two media

V2n/V1n = �T2/T1�M2n/M1n� ,

V2 = �V2n2 + Vt

2, Vt = V1 sin � ,

V2/V1 =�cos2 �T2

T1�M2n

M1n�2

+ sin2 � ,

� = � − tan−1��T1

T2·

M1n

M2n· tan �� . �4�

The ratio of Mach numbers for normal incidence can beobtained using transcendental equation4

1

M1n�k − 1���2kM1n

2 − �k − 1����k − 1�M1n2 + 2�1/2

� �2kM2n2 − �k − 1�

2kM1n2 − �k − 1���−1/2�

− 1 = M1n�1 −

1

M1n2 � − M2n�1 −

1

M2n2 ��T2

T1�1/2

M12 = M1n

2 + M1t2 , M1n = M1 cos �, M1t = M1 sin � ,

M22 = M2n

2 + M2t2 , M2t = M1t

�T1/T2, �5�

For a nonsharp interface, the Mach number does notchange across the boundary, so the ratio of normal compo-nents of the velocities is dependent on the temperature ratioonly. This eliminates the solution of the transcendental equa-tion for finding M2n, and slightly simplifies the Eqs. �1�–�4�by correcting them with M2 /M1=1.

III. NUMERICAL RESULTS FOR SW DISPERSION ONA PLASMA SPHERE AND COMPARISON WITHTHE EXPERIMENT

The goal of the calculations below is to see how theeffect of the Y-component of the transmitted velocity �therefraction effect� influences the shape of the shock front dis-persed on an interface. For comparison, we present curvesobtained using a 2D approach, first accounting and then notaccounting for the refraction effect in the case of a sharpboundary. We also made calculations for the case of a non-

sharp boundary, when the dispersion effect becomes stron-ger. In this case the angle � also gets larger, so the refractioneffect is expected to have a greater influence on the curves.

We apply our calculations to the conditions of the ex-periment described in Ref. 12. The scheme for these experi-mental results also can be found in Ref. 7 where we used it tovalidate our model. In this experiment, a spherically shapedplasma region was generated in air by focusing a laser beamwith pulse duration of 20 ns. Planar SWs generated by anelectrically exploded dielectric film were incident on theplasma region. The incident SW’s Mach number used inthese calculations was taken to be M1=1.9, which corre-sponds to the range of the experimental values. We also in-tentionally used highest temperature achieved in the experi-ment for the plasma �T2=3000 K�, and a moderatetemperature for the surrounding gas �T1=300 K� to obtain amore pronounced effect. The adiabatic coefficient in the Eqs.�5� was k=1.4.

The results for the change in the SW front as it crossesthe spherical interface between the cold gas and the hotplasma are presented in the Fig. 3. The presented curvescorrespond to the instant of time when the shock front trav-eled in the cold gas a distance equal to the half of the radiusof the sphere.

As it is seen from the figure, the correction for the re-fraction effect, as it is defined in the introduction to thispaper, makes a principal difference in the curve shapes. In-stead of spherical, the front is now bent to the opposite sidedisplaying the presence of the inflection points on its surface�2D sharp refraction corrected and 2D sharp refraction non-corrected curves, pink squares, and yellow triangles, respec-tively�. We also present a curve obtained using a one-dimensional �1D� approach �using 1D version of formula �5�for M2�, which does not account for the refraction effect. The1D and 2D curves which do not account for the refractioneffect show rather slight differences, even though consider-ing that the temperature step used in the calculations is ratherhigh.

For a nonsharp temperature change across the boundary�finite gradient�, the stronger effect of SW dispersion is eas-ily seen in the more stretched shock front �2D sharp and 2Dnonsharp, both refraction corrected, pink squares, and purplecircles, respectively� and the remarkably different shape of

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Non-sharp boundary, 2D,Refractioncorrected

Sharp,2D, NoRefraction

Sharp,1D, NoRefraction

Sharp,2D, Refraction corrected

Plasma boundary

FIG. 3. �Color online� SW dispersion on a sharp and nonsharp boundary.M =1.9, T2=300 K, T2 /T1=10.

123302-3 A. Markhotok and S. Popovic J. Appl. Phys. 107, 123302 �2010�

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Page 5: Refractive phenomena in the shock wave dispersion with variable gradients

the front. The refraction effect is more pronounced, so theinflection point disappears and the sign of the curvature ofthe front turns to the opposite. Thus the sign of the curvatureof the dispersed front makes a transition from positive tonegative through the curve with the inflection points. Whenat the very front points of the nonsharp interface curve, it isseen that they cross the longitudinal axis of the symmetry,effectively collapsing the front. As the interaction time in-creases, its further evolution will follow the usual scheme ofthe shock–shock interaction with following crossed fronts.Such a situation is possible if the temperature step is highenough and if the interaction time is long enough to fullydevelop the front. It should be noted that when the shockfront exits the sphere, the temperature step flips to the oppo-site, so the total vector of the transmitted velocity deflects offthe symmetry axis. Further development of the front shapewill proceed in the opposite direction, accounting for othertypes of interactions at the interface, which depend on theshape of the transmitted shock front.7

Such a collapse of the shock front shape was not ob-served in the experiment.12 The reasons could be that thetemperatures were rather overestimated, or the measuredshock shape inside the plasma region of Ref. 12 is not clearlyresolved. In principle, with better resolved pictures, a con-clusion about the actual sharpness of the interface can beimplicitly derived from the strength of the observed refrac-tion effect. In the case of Ref. 12, the refraction effect couldhave been reduced because of sharp boundaries, to the pointwhere the front did not develop enough to see the changes.

It is known that the sign of the temperature step acrosstwo media �slow-fast or fast-slow� and the sort of gases in-volved, can cause the SW refraction on an interface to followthe so called irregular path. In this case, the formation of aprecursor as a part of a four-wave system can occur, with thefollowing distortions of the inhomogeneous volume, devel-opment of instabilities, and vortex formation. But in the caseof a finite gradient no reflections off the interface will bepresent and hence, no irregularities of the above mentionedtype should occur during the SW dispersion.

IV. CONCLUSION

The refraction effect in the SW dispersion is a directconsequence of the SW dispersion on an interface. It happens

on both sharp and finite-gradient interfaces. The refractionphenomena in the SW dispersion consist of the deflection ofthe total transmitted velocity component toward the longitu-dinal axis of symmetry or off this axis, depending on the signin the temperature difference across the boundary. This effectrepresents a direct analogy to the optical refraction of a lightwave on a boundary between two media in which lightpropagation velocities differ, though the nature of the phe-nomena is different.

The finite-gradient boundary makes the dispersion effectstronger and consequently the refraction effect too. The cal-culations become even easier, mostly because of no need tosolve for the transmitted Mach number M2 in the transcen-dental equation. Thus the equations for the SW front shapecan be derived explicitly to its final expressions.

Comparative calculations show significant difference inthe shock front shapes if for the refraction effect is consid-ered. The SW refraction leads to variable effects includingnot only the stretching of the front surface but also its col-lapse or spreading off the axis.

The findings of this article could remarkably improve thepossibility of controlling the SW dispersion. Accounting forthe refraction effect can significantly improve the predictionsfor the shock front shapes, as well as allow getting indirectinformation about the degree of sharpness of a boundary inexperiments.

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123302-4 A. Markhotok and S. Popovic J. Appl. Phys. 107, 123302 �2010�

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