Shock Waves (2007) 17:225–239 DOI 10.1007/s00193-007-0112-z

ORIGINAL ARTICLE

Plasma mitigation of shock wave: experiments and theory Spencer P. Kuo

Received: 6 July 2007 / Revised: 17 September 2007 / Accepted: 26 September 2007 / Published online: 26 October 2007 © Springer-Verlag 2007

Abstract Two types of plasma spikes, generated by on-board 60 Hz periodic and pulsed dc electric discharges in front of two slightly different wind tunnel models, were used to demonstrate the non-thermal plasma techniques for shock wave mitigation. The experiments were conducted in a Mach 2.5 wind tunnel. (1) In the periodic discharge case, the results show a transformation of the shock from a welldefined attached shock into a highly curved shock structure, which has increased shock angle and also appears in diffused form. As shown in a sequence with increasing discharge intensity, the shock in front of the model moves upstream to become detached with increasing standoff distance from the model and is eliminated near the peak of the discharge. The power measurements exclude the heating effect as a possible cause of the observed shock wave modification. A theory using a cone model as the shock wave generator is presented to explain the observed plasma effect on shock wave. The analysis shows that the plasma generated in front of the model can effectively deflect the incoming flow; such a flow deflection modifies the structure of the shock wave generated by the cone model, as shown by the numerical results, from a conic shape to a curved one. The shock front moves upstream with a larger shock angle, matching well with that observed in the experiment. (2) In the pulsed dc discharge case, hollow cone-shaped plasma that envelops the physical spike of a truncated cone model is produced in the discharge; consequently, the original bow shock is modified to a conical shock, equivalent to reinstating the model into a perCommunicated by K. Takayama. S. P. Kuo (B) Department of Electrical & Computer Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA e-mail: [email protected]

fect cone and to increase the body aspect ratio by 70%. A significant wave drag reduction in each discharge is inferred from the pressure measurements; at the discharge maximum, the pressure on the frontal surface of the body decreases by more than 30%, the pressure on the cone surface increases by about 5%, whereas the pressure on the cylinder surface remains unchanged. The energy saving from drag reduction is estimated to make up two-thirds of the energy consumed in the electric discharge for the plasma generation. The measurements also show that the plasma effect on the shock structure lasts much longer than the discharge period. Keywords Shock wave mitigation · Plasma spike · Drag reduction · Non-thermal plasma effect PACS 52. Physics of Plasmas and electric discharges · 52.30. Ex Two fluid and multi fluid plasmas · 52.35. Tc Shock waves and discontinuities · 47.40. Ki Supersonic and hypersonic flows List of symbols D, Db L , L e, l pt , ps , pe Pt , Ps , P0 θc , θ


diameters of truncated cone frontal and rear bases heights of truncated cone and equivalent conical body, and length of spike/electrode pressure taps located at frontal side of the truncated cone, on the cone and cylinder surfaces, respectively static pressures measured by the pressure taps pt and ps , and the stagnation pressure cone half angle, upstream flow angle with respect to the z (cone’s) axis

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M, δ, β V , P, T ve,i r, z, ξ R, z 0

E, E cr νa , νi , νc νen , νin n e,i , n n

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Mach number, flow deflection angle, shock angle neutral flow velocity/pressure/temperature electron/ion fluid velocity radial and axial coordinates in the cylindrical coordinate system, r/z0 the distance [= (z 2 + r 2 )1/2 ] of a point away from the tip, effective length of the plasma spike applied electric field, breakdown threshold field attachment rate, ionization rate, ion–neutral charge transfer collision frequency electron–neutral collision frequency, ion– neutral collision frequency electron/ion density, neutral density

mance of a physical spike. Furthermore, another drawback of physical spikes is its sensitivity to off-design operation of the vehicle, i.e., flight Mach number and vehicle angle of attack. If the aspect ratio l/D of the spike length l to the aircraft frontal diameter D is less than one, it also limits the practical use of the physical spike alone for shock wave modification. Therefore, the development of new technologies for the attenuation or ideal elimination of shock wave formation around a supersonic vehicle has attracted considerable attention. The anticipated results of reduced fuel consumption and having smaller propulsion system requirements, for the same cruise speed, will lead to the obvious commercial gains that include larger payloads at smaller take-off gross weights and broadband shock noise suppression during supersonic flight. These gains can make commercial supersonic flight a reality for the average traveler. 2 Methods for flow control

1 Introduction When the spacecraft flies much faster than the sound speed (∼1,200 km/h), the airflow disturbances deflected forward from the spacecraft cannot get away from the spacecraft and form a shock wave in front of it. Shock wave appears in the form of a steep pressure gradient. It introduces a discontinuity in the flow properties at the shock front location, where the reachable edge of the reflected flow perturbates from the spacecraft. The background pressure behind the shock front increases considerably, leading to significant enhancement of the flow drag and friction on the spacecraft. Shock waves have been a detriment for the development of supersonic aircrafts, which have to overcome high wave drag and surface heating from additional friction. The design for high-speed aircraft tends to choose slender shapes to reduce the drag and cooling requirements. Although this profile is adequate for fighter planes and missiles, it is an engineering tradeoff between volumetric and fuel consumption efficiencies, and this tradeoff significantly increases the operating cost of commercial supersonic aircraft, which is preferred to be wide-body capable of carrying hundreds of people. Moreover, shock wave produces notorious sonic boom on the ground. It occurs when flight conditions (such as the altitude, speed) are changing to cause shock wave unsteady. The faster the aircraft flies, the larger the boom. The noise issue raises environmental concerns, which have precluded for example, the Concorde supersonic jetliner from flying overland. A physical spike [1] is currently used in a supersonic spacecraft to move original bow shock upstream from the blunt-body nose location to its tip location in the new form of a conical oblique shock. It improves the body aspect ratio of a blunt-body and significantly reduces the wave drag. However, the additional frictional drag occurring on the spike structure and related cooling requirements limit the perfor-

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Considerable theoretical and experimental efforts have been devoted to the understanding of shock waves in supersonic/ hypersonic flows. Various approaches to develop wave dragreduction technologies have been explored since the beginning of high-speed aerodynamics. In the following, a few of those are discussed. Busemann [2] suggested that geometrical destructive interference of shock waves and expansion waves from two different bodies could work to reduce the wave drag. However, the interference approach is effective only for one Mach number and one angle-of-attack, which make the design for practical implementation difficult. Using electromagnetic forces for the boundary layer flow control have been suggested as a possible means to ease the negative effect of shock wave formation upon flight [3]. However, an ionized component in the flow has to be generated so that the fluid motion can be controlled by, for instance, an introduced j × B force density, where j and B are the applied current density and magnetic flux density in the flow, respectively. Thermal energy deposition in front of a flying body to perturb the incoming flow and shock wave formation has been studied [4–6]. The heating of the supersonic incoming flow results in a local reduction of the Mach number. This weakens shock wave and increases the shock angle (i.e., moving the shock front upstream). Although this heating approach is effective to reduce the wave drag and shock noise in supersonic and hypersonic flows, it requires a large heating power to significantly elevate the gas temperature [6]. In other words, the energy gain from drag reduction will be much less than the consumed heating energy. Thus, this is not a practical approach for drag reduction purposes, but relatively it can be an easy approach for sonic boom attenuation [4].

Plasma mitigation of shock wave

Direct energy approaches have been applied to explore the non-thermal/non-local effects on shock waves. Katzen and Kaattari [7] investigated aerodynamic effects arising from gas injection from the subsonic region of the shock layer around a blunt body in a hypersonic flow. In one particular case, when helium was injected at supersonic speed, the injected flow penetrated the central area of the bow shock front, modifying the shock front in that area to be of conical shape, with the vertex at much farther distance from the body (at about one body diameter). Laser pulses [8,9] could easily deposit energy in front of a flying object. However, plasma generated at a focal point in front of the model had a bow radius much smaller than the size of the shock layer around the model, and its non-local effect on the flow was found insignificant. Plasmas have long been recognized to introduce non-thermal modification effects on the structure of shock waves as evidenced in a number of shock-tube experiments. However, the main inspiration for the study of plasma effects on shock waves is attributed to the observation of a wind tunnel experiment conducted by Gordeev et al. [10]. Highpressure metal vapor (high Z ) plasma, produced inside the chamber of a cone-cylinder model by exploding wire off electrical short circuit, was injected into the supersonic flow through a nozzle. A significant drag reduction was measured, which was too large to be accounted for by the thermal effect alone. In the subsequent wind tunnel experiments [11–16], the results showed that the shock front increased dispersion in its structure and/or standoff distance from the model when plasma was generated ahead of a model either by off-board/on-board electric discharges. Baryshnikov et al. [11] and Bivolaru and Kuo [15,16] investigated the relaxation time of the shock structure modification in decaying discharge plasma. A long-lasting plasma effect on the shock structure (i.e., after the discharge ceases, it takes much longer than the discharge period to recover to the baseline state) was observed in both experiments. Flow deflection by plasma approaches is studied in the present work. This is realized that the shock wave angle β and the shock structure depend on the Mach number M and the flow deflection angle θ [17], and the charged parti-

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cles in plasma, accelerated by the applied electric field, can deflect the flow through collisions. The deflection is particularly effective when plasma is produced directly within the neutral flow. Ions moving through their own gas are subject to charge transfer between the ion and the neutral gas, a type of inelastic collision whose probability predominates over those of other interactions in the low ion energy regime [18]. The charge transfer cross-section between N+ 2 and N2 in the relevant energy regime is larger than 3 × 10−19 m2 . An ion that has traveled a single charge-transfer free path becomes a neutral particle but retains its velocity, which is usually low. Most of the converted neutrals move at subsonic speed; these particles do not contribute to the shock wave formation. Some of them may move at supersonic speed; however, they do not drift together, and thus the produced disturbances off the model are expected to spread out in the form of expansion waves, rather than coalesce into shocks. The ions converted from neutrals through charge-transfer are collected by the cathode and do not contribute to the shock wave generation either. Electrons deflect the neutral flow through elastic collisions. The shock of the deflected flow is expected to have a larger shock angle (than that of the baseline one) and a modified structure, representing a weaker shock.

3 Experimental setup Experiments were conducted in the test section, with a 0.38 m × 0.38 m cross section, of a supersonic blow-down wind tunnel; a schematic of the wind tunnel is presented in Fig. 1a. The upstream airflow had a flow speed v = 570 m/s, temperature T1 = 135 K, and a pressure P1 = 0.175 atm. 3.1 Wind tunnel models Both of the wind tunnel models used in periodic and pulsed discharge operations have a similar truncated-cone body connected to the same cylindrical body attached to a holder. In addition, each model also consists of a sharpened solid tungsten rod of a diameter d = 2.4 mm, and a ceramic insulator which holds the tungsten rod in place concentrically with

Fig. 1 Schematics of a the supersonic wind tunnel facility used in the experiments, b the model used in periodic electric discharge operation, and c the model used in pulsed dc discharge operation

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Fig. 2 Schematics of the circuits for a the 60 Hz periodic electric discharge and b the pulsed dc electric discharge; a voltage divider and a current probe are shown in both circuits. c Power function of a dc pulsed

electrical discharge during a wind tunnel run. The airflow is supersonic (Mach 2.5) in the time interval from t = 2.3 to 7.6 s

the truncated-cone body to form the electrodes for the discharge. The schematics of the two models are presented in Fig. 1b and c. The truncated 60◦ cone has a frontal diameter D = 11.1 mm and a height L = 12.7 mm. The cylindrical base of the cone has a diameter Db = 25.4 mm. The gap between the tungsten rod and the inner wall at the front of the truncated-cone body is 3.5 mm. The breakdown voltage is provided by two types of power supply, one for 60 Hz periodic electric discharge and the other one for pulsed dc discharge. The noses of the models used in the two cases have different configurations. The nose of the model shown in Fig. 1b, used in periodic discharges, is a cone-shaped ceramic insulator with a short protruding spike, which replaces the truncated part of the cone. The distance from the tip to the edge of the truncated-cone surface is about 5 mm. On the other hand, the nose of the model in Fig. 1c, used in pulsed dc discharges, has a (l =) 9 mm spike out of the center of the flat circular cross section, protruding to the tip location of a perfect cone. Three pressure taps (labeled by pt , ps , pe ) are installed in this model. One ( pt ) at the frontal side of the truncated cone is located in the gap between the ceramic insulator and the spike to directly measure the frontal total pressure on the model. The other two ( ps and pe ) are placed on the cone and cylinder surfaces at locations behind the discharge to detect the spatial extent of the plasma effect on the flow field. These two static pressure taps are made by drilling holes on the cone and cylinder surfaces at z = 19.2 mm and z = 27.5 mm, along the cone–cylinder axis away from the tip (located at z = 0) of the spike. All the taps are connected through adequate tubing (less than 0.6 m long) to high-speed piezoelectric transducers (83 µs response time) located outside the wind tunnel.

voltage across the electrodes is low and there is no discharge. During the other half cycle, the diode has a reverse bias. The charged capacitor increases the voltage across the electrodes and breakdown occurs. The discharge normally initiates in the region near the tip electrode where the applied electric field concentrates due to the cylindrical geometry. Therefore, it prefers the tip electrode to be negative so that the electrons can be pushed to the upstream region. Indeed, plasma effect on the shock wave structure has been observed only when the tip of the model is designated as the cathode. The voltage across the electrodes has an asymmetric waveform with a peak of about 4.5 kV, exceeding the 4 kV (for 5-mm gap) required for avalanche breakdown. The electric field intensity near the tip exceeds 1 MV/m before breakdown occurs. It reduces to be less than 100 kV/m as the discharge current reaches the peak. The peak and average power of the discharge are about 1.2 kW and 100 W, respectively. The plasma density and temperature of the discharge were not measured in the runs. However, the electrode arrangement and the power supply were similar to those used in producing 60 Hz plasma torch, which was measured [19] with the peak electron density exceeding 1019 electrons/m3 . During the run, the background pressure dropped. It reduced the gas breakdown threshold field, thus the ionization rate increased and the plasma density should increase.

3.2 Periodic discharge The 60 Hz power supply for periodic discharge operation is shown in Fig. 2a. During one of the two half cycles when the diode is forward biased, the capacitor is charged so the

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3.3 Pulsed DC discharge The power-supply used for pulsed dc discharge is shown in Fig. 2b. The charging and discharging time of the RC circuits are about 1 s and 20 ms, respectively. It produces energetic plasma with a repetition rate less than 1 s−1 . The discharge current increases rapidly to about 20 A shortly following the gaseous breakdown [15,16]. Hence, the discharge turns to the (diffused) arc mode, and the voltage across the electrodes (with a gap of 3.5 mm) drops rapidly from −3.2 kV to about −200 V. The capacitor bank cannot maintain the arc

Plasma mitigation of shock wave

discharge, then the current drops to zero. Shown in Fig. 2c is the power function of a discharge in the supersonic airflow, the discharge lasts about 15 ms. The peak power exceeds 40 kW, and the energy in this pulse is about εe1 = 150 J. In each discharge much of the energy stored in the capacitor bank is dissipated in the 150  resistor in series with the discharge, and lost to the internal resistance of the capacitor. Based on the results of the voltage and current measurements, the energy distribution is calculated to be: (1) 760 J stored initially in the capacitor; (2) 500 J lost to 150  ballasting resistor; (3) 150 J delivered to the discharge; (4) 80 J left in the capacitor after the main discharge; and (5) 30 J lost in the capacitor. 3.4 Optical diagnostics Shadowgraph/Schlieren methods are used to optically diagnose the flowfield around the spike and nose of the cone. A black and white charge coupled device (CCD) camera, with a frame rate of 30 frames per second and exposure time of 1/60 s (which is slightly less than four times of each discharge period in the periodic operation), is used to record directly the shadowgraph/Schlieren images of the flow dynamics. A video camera as the corresponding one to the CCD camera is used to record the spatial distribution and temporal evolution of the plasma glow with the same frame rate and exposure time. The video graph recorded in each frame is an integrated result over the exposure time, and thus the temporal variation of the shock wave structure and plasma glow during a single discharge period cannot be recorded directly. Continuous video graph of the flow can reveal important information regarding the dynamic behavior of the flow field. Although the starting times in recording each event (i.e., the starting time of each frame) by the two cameras are not synchronized, the events recorded by the two cameras can still be synchronized by counting number of frames from the reference frames, except, there will be a maximum possible time difference that is half of the exposure time (i.e., 1/120 s). Therefore, the results extracted from videotapes recording the shadowgraph/Schlieren images of the flow and plume images of plasma can provide the correlation between the

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plasma distribution and the modification of the shock structure, which will be a very useful information regarding the experiment because the plasma effect on shock wave is not expected to be always observed in the experiment. Any consistent relationships appearing in the correlation can help to deduce required plasma conditions to achieve significant plasma effect on the shock wave, as being elaborated in Sect. 6.

4 Experimental results from 60 Hz periodic electric discharge The produced spray-like plasma acted as a spatially distributed spike, which could deflect the incoming flow. A video camera was used to tape the shadow video graph of the flow. The modification effect depended on the density and volume of the plasma spike produced by the discharge, which varied with time; in other words, the plasma spike increased its size and intensity from near zero to the maximum and then to decay to near zero. This time varying spike could cause the shock front position to also vary in time. Although the temporal variation of the shock wave structure during a single discharge period could not be recorded directly, the desired information regarding the transient behavior of the flow field was extracted from the continuous shadowgraphy of the flow. This is demonstrated in Fig. 3, which includes a sequence of four shadowgraphs showing the responses of the shock wave to the growth and decay of the plasma spike in a discharge cycle. In this and other shadowgraphs as well as the Schlieren images and plasma plume images presented later, the flow is from left to right. The growth of the plasma spike is manifested by the variation of the background brightness in the shadowgraphs. First shadowgraph shown in Fig. 3a is in the case that the discharge is off. Therefore, the shadowgraph is dark (absence of background light from the plasma) and is the baseline one. The next two presented in Fig. 3b and c correspond to the situation that the discharge is intensifying before reaching the peak. The baseline shock front is first split into two (Fig. 3b), with a new one located upstream; the entire baseline shock front

Fig. 3 An assembled time sequence of four shadowgraphs (a–d) to represent the flow response to the plasma spike during one discharge period in the middle of a wind tunnel run at Mach 2.5

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Fig. 4 A sequence of shadowgraphs taken during a wind tunnel run in the presence of plasma. a At the instance close to initiating plasma, b at a later time during the run, c at a distinctively later time during the

is then moved to the new one location as the plasma spike is further intensified (Fig. 3c). As the plasma spike is intensified to reach the peak, its modification effect on the shock structure also reaches the maximum. The shock front in Fig. 3d is very diffused; it spreads from the one shown in Fig. 3c to further upstream region. As the shock front moves upstream, its shock angle also increases. These observations on unsteady shock motion containing the aforementioned features of the flow field are typical of all the experiments performed and seen in the recorded videotape. The pronounced influence of plasma on the shock structure is demonstrated by the result shown in Fig. 3d. A comparison of Fig. 3d and a clearly observes a transformation of the shock from a well-defined attached shock into a diffused and highly curved shock structure having a larger shock angle. This modification is an indication of shock wave being weakened by this plasma spike. This is an encouraging result, evidencing the effectiveness of this plasma scheme in reducing wave drag at supersonic speeds. To further examine the flow structure, a Pitot tube was installed in the tunnel, which can be seen with its usual detached shock front on the top portion of the shadowgraphs presented in Fig. 4. Figure 4a is a snap shot of the flow at the instance close to initiating the plasma. As shown an undisturbed conical shock is formed in front of the plasmaproducing model. Figure 4b taken at a later time during the run, on the other hand, clearly demonstrates the pronounced influence of plasma on the shock structure. Comparison of Fig. 4a and b clearly indicates an upstream displacement of the shock front along with a larger shock angle indicating a transformation of the shock from a well defined attached shock into a classic highly curved bow shock structure. It is also interesting to note that the shock in front of Pitot probe, which is placed at a distance above the plasma-producing model, has been noticeably altered as evident from the larger shock angle. A highly diffused detached shock front is observed in Fig. 4c taken at a distinctively later time during the same run. The diffused form of the shock front could be the result of less spatial coherency in the flow perturbations introduced by the spatially distributed plasma; it could also ascribe to a visual effect from an asymmetric shock

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same run, and d at the time when the discharge is around the peak and shock wave is eliminated

Fig. 5 A superimposed shadowgraph (from the three shadowgraphs in Fig. 4a–c) showing the (attached) baseline shock front and two detached shock fronts modified by plasma with increasing discharge intensity; the insert shows the plume image of the plasma produced by the electrical discharge

front caused by the non-uniformity of the generated plasma, a well-known integration effect inherent in the shadowgraph technique when visualizing a three-dimensional flow field. This phenomenon is commonly observed when the spatial extent of the region leading to the shock is small compared to the test section dimensions which is, however, not the case of the present experiment. Closer examination of Fig. 4c demonstrates a further upstream propagation of the bow shock having an even more dispersed shape and a larger shock angle. It is also interesting to note that the shock wave in front of Pitot probe has also moved upstream and some evidence of flow expansion may be seen near the tip of the probe. This is an interesting result indicating that the effect of plasma is not confined to the vicinity of the plasma-generating model but rather influences a large region of the flow field. The shadowgraphs of the flowfield presented in Fig. 4a–c are superimposed into one presented in Fig. 5. The insert in Fig. 5 exemplifies that plasma generated by the discharge has a distribution spreading around the cone model. Clearly, plasma causes the shock front to move upstream with increasing standoff distance

Plasma mitigation of shock wave

from the model and to become more and more diffused in the process. The detached shock fronts appear more diffused than the attached one (which is also modified by the plasma) presented in Fig. 3d. A shock-free state is observed as the discharge is further intensified. This is demonstrated in Fig. 4d. Using plasma to eliminate shock is an ultimately desirable result that has significant consequences in minimizing wave drag and shock noise at supersonic speeds. Based on the measured peak and average discharge power, the upper bound of the peak and average temperature enhancements are estimated to be Tpeak ∼ = 26 K (for Pin |peak = 2.2 K (for P 1.2 kW) and Tave ∼ = in |average = 100 W). The peak temperature perturbation is less than 10% of the unperturbed, 285 K, in the region behind the shock, and less than 20% of the upstream gas temperature of 135 K; and the average temperature perturbation is less than 1 and 2%, respectively. Thus the heating could at most increase the sound speed and decrease the Mach number by 5%. These changes were too small to introduce any significant thermal effects. The deflection of the incoming flow by a symmetrically distributed plasma spike in front of the shock as a likely process responsible for the observed modifications on the shock wave structure will be modeled and analyzed in Sect. 7.

5 Experimental results from pulsed DC electric discharge The baseline shadowgraph image of the flowfield (i.e., in the absence of plasma), in front of the model shown in Fig. 1c, is

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presented in Fig. 6a. As shown, a bow shock was generated in front of the truncated cone. This slender central-spike did not cause noticeable modification on the shock wave structure, but a region with elevated pressure around the spike was formed. This high-pressure layer, with a conic shape, was extended to the surface location of the truncated part of the cone, as seen in a baseline Schlieren image presented in Fig. 6b. As plasma was introduced through a discharge at the spike region of the shock-generating cone with the spike of the model designated as the cathode, it was found that the Schlieren image of the flowfield could be quite different from that shown in Fig. 6b. The discharge could produce apparently cone-shaped plasma around the spike of the model, as demonstrated by the video graph in Fig. 6c. This plasma had an axially symmetric distribution based on the spatial distribution of the discharge glow intensity. Moreover, it was hollow because no glow was detected from the inner region of the cone-shaped plasma. Comparing the Schlieren images of the flowfield presented in Fig. 6b and d, it is found that the original bow shock in front of the nose of the model is not there any more. The experimental shock front in Fig. 6d is compared to a conical shock front evaluated numerically for a perfect cone. This comparison is presented in Fig. 7. As shown, the two shock fronts almost overlap each other. This concurrence suggests that the discharge-produced plasma shown in Fig. 6c reinstates the model to a perfect cone configuration [15,16, 20]. Such a change is equivalent to a 70% increase in the body aspect ratio, from L/Db = 0.5 (blunt body with slender attached spike with aspect ratio of l/D = 0.81) to L e /Db =

Fig. 6 a Shadowgraph and b Schlieren images of the flowfield in the absence of plasma; c cone-shaped plasma produced in front of the nose of the model with a symmetric distribution around the physical spike and d the corresponding Schlieren image of the flowfield showing strong plasma effect on shock wave

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Fig. 7 A comparison of the shock front of a plasma aero-spike with that of a 60◦ perfect cone obtained numerically

0.85 (60◦ conical body), where L e = L + l; leading to significant drag reduction to be shown in the pressure measurements. This plasma effect is characteristically different from that observed in the periodic discharge case, showing increased angle and dispersion of the shock front as well as standoff distance from the model. Since the wave drag of an oblique shock is much smaller than that of a bow shock, a reduction in the drag force on the object is expected when the cone-shaped plasma is introduced. This is verified by the pressure changes on the model surfaces, measured by two pressure-transducers placed on the front and cone surfaces, respectively. The measured pressures Pt and Ps (normalized to the stagnation pressure P0 ) at the respective locations as functions of time are plotted in Fig. 8. In 4 s time interval from t =

Fig. 8 Pressures measured through pressure taps placed a at the front ( pt ) and b on the cone surface ( ps ) of the model

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3 to 7, wind tunnel was running in steady state supersonic (Mach 2.5) mode, and three discharge events manifested by the spikes were recorded (overall there were five discharges from 2 to 8). It is noted that the recorded data is an average value over the time delay period introduced by the tubing connecting the pressure top and the transducer. However, it should provide reliable results on the integrated pressure change over the perturbation period. In the following, the discharge event around t = 4 is chosen for analysis and discussion. The unperturbed pressures, Pt0 and Ps0 , at the two locations were measured to be about 1.86 × 105 and 0.977 × 105 N/m2 , respectively. These readings were changed by the discharge. The pressure function (Pt /Po ), at the front location, is presented in Fig. 8a. As shown, the pressure dropped considerably and this lasted (∼80 ms) much longer than the period (νin ) is the ion–neutral charge transfer collision frequency, which is dominated by charge transferring between the same type particles. Thus, after the charge transfer, the ion gains the neutral’s initial velocity V10 , and the neutral’s velocity reduces to the ion’s initial velocity. Since the ion’s velocity is low, the converted neutrals form a subsonic flow, which will not contribute to the shock wave formation. On the other hand, the converted ions form a supersonic flow; however, this ion flow will be collected by the cathode to close the discharge current loop, and will also not contribute to the shock wave formation. Therefore, each of the ion and neutral fluids can be decomposed into two components, i.e., (1) (2) (1) (2) n i vi = n i1 vi + n i2 vi and n n V1 = n n1 V1 + n n2 V1 . The one with superscript 1 is related to shock formation, and the other, with superscript 2, is not related to shock formation. dn i2 /dt = dn n2 /dt = νc n i1 , and n n2  n n1 because the interaction region (plasma layer) is very narrow. To simplify the analysis, the electron–ion collision terms in (1) and (2) will be neglected. Thus (1) to (3) become   (1) (4) m e d(n e ve )/dt = −n e m e νen ve − V1 − en e E     (1) (1) (1) (1) m i d n i1 vi /dt = −n i1 m i νin vi −V1 − n i1 m i νc vi

m e d(n e ve )/dt = −n e m e νen (ve − V1 ) + n e m e νei (vi − ve )

where αn = [νc /(νin + νc )](eE 0 tn /V10 m n )(n e0 /n n ) and βn = (n e0 /n n )(νc tn )[1 + νin /(νin + νc )]. Thus the deflected flow has spatially dependent deflection angle θ
and Mach number M1 , which are obtained from (8) to be θ
(r ) = tan−1 [V1r (r )/V1z (r )] and M1 (r ) = {[V1r (r )2 + V1z (r )2 ]1/2 / V10 }M10 , where M10 is the Mach number of the unperturbed flow.

−en e E

(1)

m i d(n i vi )/dt = −n i m i νin (vi − V1 ) − n i m i νc (vi − V10 ) −n e m e νei (vi − ve ) + en i E

(2)

m n d(n n V1 )/dt = n e m e νen (ve − V1 ) + n i m i νin (vi − V1 ) +n i m i νc (vi − V10 )

(3)

+en i1 E (5)     (1) (1) m n d n n1 V1 /dt = n e m e νen ve − V1 + n i1 m i νin   × vi(1) − V1(1) − n i1 m i νc V10 (6) Neglecting the inertial terms on the left-hand side of (4) and (5), we obtain n e m e νen (ve − V1(1) ) ∼ = −n e eE and (1) (1) (1) [ν /(ν + ν )]n (eE − m i νc V1 ); n i1 m i νin (vi − V1 ) ∼ = in in c i1 thus (6) reduces to dV1(1) /dt = −[νc /(νin + νc )](n e /n n )eE/m n − (n i1 /n n )νc   × V10 + νin V1(1) /(νin + νc ) (7) where n i1 ∼ = n e is assumed. It is noted that in (3) the pressure gradient term is neglected by assuming that the density and temperature of the airflow do not change considerably during the transit period of the airflow passing through the plasma spike. We now integrate (7) over a transit period tn = z 0 /V10 , the time for the airflow to pass through the plasma spike of length z 0 . We obtain (1) V1 (r, tn ) ∼ = V10 zˆ − V10 exp{−η[1 − (1 + ξ 2 )−2.65 ]} ×[αn (1 + ξ 2 )−1 (ˆz − ξ rˆ ) + βn zˆ ]

(8)

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7.2 Supersonic flow over a cone We now apply Taylor–Maccoll’s theory to analyze the deflected supersonic flow (8) over a cone. We start with a simple situation that the incoming flow from the left propagates at a constant angle θ
with respect to the axis of the cone. In the steady state, a conic shock front signified by a step pressure jump is formed to separate the flow into regions 1 and 2 of distinct entropies as sketched in Fig. 12. In the figure, the cone is placed horizontally (along the z-axis); thus, the flow velocity V1 in region 1 has an angle θ
with respect to the z (cone’s) axis; the flow has a Mach number M1 . The geometry of Fig. 12 is adapted from Fig. 10.4 in the book by Anderson [17], which is for the special case with θ
= 0. The conic shock has an angle β to be determined. In region 2 immediately behind the shock front, the flow has a deflection angle δ with Mach number M2 and velocity V2 = VR2 aˆ R − Vθ2 aˆ θ , where aˆ R and aˆ θ are unit vectors in spherical coordinate system, the origin is at the tip of the cone, and the z axis is along the cone axis. The shock angle β and deflection angle δ are related by the δ − β − M relation of a wedge, which is derived through the continuity conditions at the flow discontinuity and in the case of θ
= 0 can be found in textbooks, as, for example, Eq. 4.17 in the book by Anderson [17]. This relation can easily be generalized for θ
= 0. Because the changes across a conic shock wave (similar to across an oblique shock) are governed by the normal component of the free-stream velocity, the relevant parameters in the equations are Mn1 = M1 sin(β − θ
) and Mn2 = M2 sin(β − δ). Letting β
= (β − θ
) and δ
= (δ − θ
), these two relations become Mn1 = M1 sin β
and Mn2 = M2 sin(β
−δ
), which are expressions similar to those in the θ
= 0 case. Therefore, the δ
−β
− M relation (similar to that for δ − β − M [17,22]) is derived to be

Fig. 12 Geometry for the numerical solution of deflected flow over a cone

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S. P. Kuo

tan δ
= 2 cot β




   M12 sin2 β
−1 / M12 (γ +cos 2β
) + 2 (9)

where γ is usually chosen to be 1.4. The normalized Taylor–Maccoll equation for conical flows (Eq. 10.15 of Anderson [17]) is expressed as 0.2[1 − G 2 −G
2 ][2G +G
cot θ + G

]−G
2 [G +G

] = 0 (10) where G = VR2 /V2 max , G
= dG/dθ, G

= d2 G/dθ 2 , and γ = 1.4 is assumed. The boundary conditions of (10) are given by G(β) = f(M2 ) cos(β − δ) and G
(β) =−f(M2 ) sin(β −δ) (11) where f(M2 ) = V2 /V2 max = [(5/M22 ) + 1]−1/2 , M2 = Mn2 / sin(β−δ), Mn2 = {[(M1 sin β
)2 +5]/[7(M1 sin β
)2 − 1]}1/2 , and δ is determined by (9). Equation (10) will be solved based on the numerical procedure outlined in Sect. 10.4 of Chap. 10 in the Anderson book [17]. That is, a proper β = βc is found by iteration in solving (10) such that the normal component of the flow veloc
(θ ) = G
(θ )V ity on the cone surface Vθ2 (θc ) = VR2 c c 2 max becomes zero, where θc is the half cone angle of the model. The effect of a localized plasma spike on the shock wave is then inferred from changes in the deflection angle θ
and in the Mach number M1 of the flow, where θ
and M1 vary with r , the radial coordinate with respect to the z-axis. 7.3 Numerical results The deflection angle θ
(r ) and the Mach number M1 (r ) of the deflected flow vary with the intensity of the discharge (gauged by the peak electron density n 0 ), where M10 = 2.5 was used in the numerical calculations. For tn = 0.88 × 10−5 s (i.e., z 0 = 5 mm), n n ∼ 1025 m−3 (i.e., P1 = 0.175 atm and T1 = 135 K), V10 = 570 m/s, νc ∼ = 2νin ∼ = 2× 109 s−1 , and assuming E 0 ∼ 106 V/m, we then have αn = 0.309 × 10−14 n e0 and βn = 0.235 × 10−14 n e0 . Let n e0 = 3.24 × 1020 (1 − ζ )m−3 , so that αn = (1 − ζ ) and βn = 0.76(1 − ζ ). The two functions θ
(r ) and M1 (r ) are plotted in Fig. 13a and b for the parameter ζ = 0.77, 0.8, and 0.9. Using these results for each ζ (i.e., n 0 ) as the parameters at the shock front location, the corresponding oblique angle βc (r ) = βc
+ θ
of the shock front can be determined by solving Eq. (10) iteratively to meet the condition that the normal component of the flow velocity on the cone surface G
(θc ) = 0. Thus the position of the shock front can be determined by the trajectory equation dz/dr = cot βc = cot(βc
+ θ
)

(12)

Plasma mitigation of shock wave

237

Fig. 13 a θ
(ξ ), the deflection angle, and b M1 (ξ ), the Mach number, of an incoming flow after being scattered by a plasma spike; c attached shocks in a supersonic flow over a 60◦ cone (represented by the shadow region) for two cases, ζ = 1 (no discharge) and ζ = 0.8 (corresponding to an intense discharge), where ξ = r/z 0 and η = z/z 0 . The line labeled ζ = 1 represents the baseline shock front. The insert is a superimposed shadowgraph showing a baseline shock front and a modified shock front by the plasma, for comparison with the numerical results

The result in the case of ζ = 0.8 (corresponding to a relatively intense discharge) is presented in Fig. 13c, in which the baseline shock front, having a shock angle β0 = 42.6◦ (for θc = 30◦ ), is also presented for comparison. As shown, the shock angle is increased to 46◦ by the plasma spike; agreeing well with the experimental result which is inserted in the same figure for comparison. In this case, the peak electron density of the plasma spike used in the numerical calculation is n 0 = 6.5 × 1019 m−3 , which agrees with that produced by the on-board diffused arc discharge in the experiments. For each ζ , a βc (ξ ) distribution is determined. In terms of the determined βc (ξ ) and δ for β
and δ
in (9), one can obtain an equivalent Mach number distribution M1eq (ξ ). This is the Mach number distribution for an undeflected flow (i.e., in the absence of the plasma spike) to generate the same shock structure as that in a plasma-deflected flow over the same cone. It is found that the effective Mach number M1eq (ξ ) of the incoming flow in the tip region has a similar spatial distribution as the corresponding M1 (ξ ) presented in Fig. 13b, and is smaller than M10 . 7.4 Cone-shaped plasma In the pulsed dc discharge case, there are two factors that affect the plasma shape. One is the length l of the added spike, which has to satisfy the conic relation Db /2(l+L) = tan 30◦ . The other one is the high-pressure region formed around the spike (e.g., shown in Fig. 6b). This region had a higher neutral density, which increases the breakdown threshold field, thus a surface flash discharge along the boundary surface of this high pressure region was initiated. Consequently, the produced plasma had a hollow conical shape, explaining the experimental observation. The ion–neutral charge transfer process was active [18]. Instead of blowing the ions out of the discharge region through the elastic collisions, part of

the neutral flow converted into ion flow through the charge transfer process. These ions were held in the discharge region by the applied electric field; they were collected by the spikecathode and did not contribute to the shock wave generation. The converted neutrals moved at ion speed that was subsonic; thus these particles did not contribute to the shock wave formation either. Electrons interacted with the neutral flow through elastic collisions and were held in place by the space charge field of ions. The momentum transfer rate from electrons to neutrals was larger than that from ions to neutrals [24], and electron–neutral elastic collisions symmetrically deflected the incoming flow. Consequently, the plasma was balanced in a conical shape, and the shock structure was also modified to a conic shape. 8 Summary and concluding comments Wind tunnel experiments were conducted to explore the nonthermal plasma effects on the shock wave structures. Two imperfect cone-shaped models were used as the shock generator and were facilitated with electrodes for on-board discharges to generate plasmas in front of the models. One model was operated with 60 Hz periodic discharge and the other one with pulsed dc discharge. The tip of the central electrode in both models was shaped to match the cone angle. It turned out that the sharp tip also worked to enhance the electric field intensity in the region in front of the tip. Moreover, the central electrode of each model was arranged as the cathode. This arrangement together with the favorable electric field distribution made electron current in the discharge much easier to pass through the shock front into the upstream (lower pressure) region before returning to the body of the model as the anode. As shown in Figs. 9c and 10c, the discharges could produce plasmas in the upstream region and with symmetric distributions around the central electrode,

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238

which were found to be the necessary conditions to achieve noticeable plasma effects on shock waves. In the periodic discharge case, the introduced plasma caused the shock front to have increased dispersion in its structure as well as its standoff distance from the model. A shock-free state (Fig. 4d) was observed as the discharge was intensified. A theory based on the deflection of the incoming flow by a symmetrically distributed plasma spike in front of the shock as the process to modifying the shock wave structure has been formulated and analyzed numerically. As demonstrated in Fig. 13c, the numerical results are in good agreement with the experimental results. The wave drag of the shock on the cone depends on the strength of the shock, which in turn depends on the Mach number of the flow. It is found that the effective Mach number M1eq (ξ ) of the deflected flow in the tip region is smaller than M10 . A decrease in the effective Mach number of the incoming flow in the tip region verifies that the plasma spike can indeed reduce the wave drag of the shock on the cone. Moreover, the modified shock structure moves upstream away from the cone; it also results to the reduction of the wave drag on the cone. It is noted that the solution of the Taylor–Maccoll equation represents an attached shock. When θc in Fig. 12 exceeds a maximum value, i.e., θc > θcmax , there exists no Taylor– Maccoll solution, and the shock becomes detached [17]. θcmax decreases with the decrease of M1 and the equivalent θc of the cone increases with the increase of θ
. As shown in Fig. 13, M1 decreases and θ
increases as ζ decreases (i.e., plasma density increases). ζ = 0.77 is the lowest value for which a Taylor–Maccoll solution can be obtained. In other words, the numerical analysis indicates that the shock becomes detached when ζ < 0.77 (i.e., n 0 > 7.5 × 1019 m−3 ). Its standoff distance from the model is expected to increase with a further increase in n 0 , consistent with the experimental results presented in Fig. 5. This theory facilitates the understanding of the experimental observations, which cannot be reasonably explained by a simple heating effect. In the pulsed dc discharge case, it was found that only when the spike length l satisfied the conic relation Db /2(l + L) = tan 30◦ , the produced plasma had a hollow conical shape and a favorable aerodynamic change from a bow shock of the truncated cone to a tip-attached conical shock. The pressure measurements showed that the introduced plasma reduced the wave drag on the model considerably. Drag reduction made up about two-thirds of the extra energy consumed in the electric discharge for the plasma generation. The electric power pulse length in Fig. 2c is less than 10 ms, whereas the pressure pulse lengths in Fig. 8 are longer than 80 ms. This long-lasting plasma effect on the shock structure is ascribed to the non-thermal mechanism, which improves the energy efficacy in performing shock mitigation. The experimental results demonstrate that the blunt body aero-

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dynamics can be greatly improved by introducing a properly shaped plasma spike in front of it. Acknowledgements The author would like to thank Dr. Daniel Bivolaru and Prof. I. M. Kalkhoran for their contribution to the experiments, to thank Dr. Steven Kuo for helping with the numerical work, and to acknowledge one of the referees for constructive comments helping to improve the manuscript. Work was supported in part by the Air Force Office of Scientific Research Grant AFOSR-FA9550-04-1-0352.

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Plasma mitigation of shock wave 21. Kuo, S.P.: Conditions and a physical mechanism for plasma mitigation of shock wave in a supersonic flow. Physica Scripta 70, 161– 165 (2004) 22. Kuo, S.P., Kuo, S.S.: A physical mechanism of non-thermal plasma effect on shock wave. Phys. Plasmas 12, 012315(1–5) (2005)

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