Plasma Mitigation of Shock Waves ⎯ Experiments and Theory S. P. Kuo Department of Electrical & Computer Engineering Polytechnic University Six Metrotech Center, Brooklyn, NY 11201 Email: [email protected] Abstract. Experiments were conducted in a Mach 2.5 wind tunnel to explore the modification effect on the shock wave structure by a plasma spike generated by an onboard 60 Hz electric discharge in front of a 600 cone-shaped model, which was used as a shock wave generator. The pronounced influence of plasma on the shock structure is demonstrated by the experimental results, showing a transformation of the shock from a well-defined attached shock into a highly curved shock structure, which has increased shock angle and also appears in diffused form. Due to cyclic nature of the generated plasma an unsteady shock motion during one discharge period was observed. 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. Experimental results exclude the heating effect as a possible cause of the observed shock wave modification. A theory also using a cone model as the shock wave generator is presented to explain the observed plasma effect on the shock wave. Analysis shows that the plasma spike can effectively deflect the incoming flow before the flow reaches the cone model; such a flow deflection modifies the structure of the shock wave generated by the cone model from conic shape to a curved one. The shock front moves upstream with a larger shock angle, consistent with the experimental results.

Contents 1. 2. 3. 4. 5. 6.

Introduction 2 Methods for the Flow Control 2 Plasma Spikes for the Mitigation of Shock Waves 4 Experimental Setup 5 Experimental Results 7 Theory and Results 11 6.1. Supersonic flow over a cone . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2. Plasma spike and flow deflection . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7. Concluding Comments 17 Acknowledgement 18 References 18

1.

Introduction

A flying object agitates the background air; the produced disturbances propagate, through molecule collisions, at the speed of sound. When the object flights approaching to the sound speed (roughly 1200 km/h at level flight), those disturbances deflected forward from the object move too slow to get away from the object and form a sound barrier in front of the flying object. Ever since Chuck Yeager and his Bell X-1 first broke the sound barrier in 1947, aircraft designers have dreamed of building a passenger airplane that is supersonic, fuel efficient and economical. However, the agitated flow disturbances by the flying object at supersonic/hypersonic speed coalesce into a shock appearing in front of the object. 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 is the reachable edge of the flow perturbations by the object. The background pressure behind the shock front increases considerably, leading to significant enhancement of the flow drag and friction on the object. 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. While that profile is fine for fighter planes and missiles, it has long dampened dreams to build a wide-bodied airplane capable of carrying hundreds of people at speeds exceeding 1200 km/h. This is an engineering tradeoff between volumetric and fuel consumption efficiencies and this tradeoff significantly increases the operating cost of commercial supersonic aircraft. Moreover, shock wave produces notorious sonic boom on the ground. It occurs when flight conditions 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 spike1 is currently used in the supersonic object 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 performance of a physical spike. Also 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 the Flow Control

Considerable theoretical and experimental efforts have been devoted to the understanding of shock waves in supersonic/hypersonic flows. Various approaches to develop wave drag-

2

reduction technologies have been explored since the beginning of high-speed aerodynamics. In the following, a few of those are discussed. Buseman2 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 possible means to ease the negative effect of shock wave formation upon flight3. 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 x B force density, where j and B are the applied current density and magnetic flux density in the flow. Thermal energy deposition in front of the flying body to perturb the incoming flow and shock wave formation has been studied4-6. Heating of the supersonic incoming flow results in a local reduction of the Mach number. This weakens the shock wave by increasing the shock angle (i.e., moving the shock front upstream). Although this heating effect is an effective means of reducing the wave drag and shock noise in supersonic and hypersonic flows, it requires a large power density to significantly elevate the gas temperature6. It is known that use of the thermal effect to achieve drag reduction in supersonic flight does not lead to energy gain in the overall process. Thus, this is not a practical approach for drag reduction purposes, but it can be a relatively easy approach for sonic boom attenuation4. Direct energy approaches have been applied to explore the non-thermal/non-local effects on shock waves. Katzen and Kaattari7 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 conical shape with the vertex at much farther distance from the body (at about one body diameter). Laser pulses8,9 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. Plasma has the potential to possibly offer a non-thermal modification effect on the structure of shock waves. The results from early and recent experiments conducted in shock tubes exhibited an increased velocity and dispersion on shock waves propagating in the glow discharge region10-12. Appartaim et al.12 further show that the plasma effect increases with an increase of the atomic weight of the plasma (generated in noble gas), unambiguously excluding the thermal effect to be responsible for the observed shock acceleration. The study of the plasma effect on shock waves was inspired by the observation of a wind tunnel experiment conducted by Gordeev et al.13. High-pressure metal vapor (high Z) plasma, produced inside the chamber of a cone-cylinder model by exploding wire off electrical short circuit, is 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. A brief history of the development of this work was reported in an article published in the Jane’s Defence Weekly14. In the other wind tunnel experiments, 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 discharges15-22 or by off-board microwave

3

pulses23. Exton et al.24 applied a seeding approach to generate plasma in front of the baseline shock front by the on-board microwave pulses, however, the plasma was too weak to introduce any visible effect on the shock wave. Baryshnikov el al.18 investigated the relaxation time of the shock structure modification in decaying discharge plasma. The observed long-lasting plasma effect on the shock structure, attributed to the existence of long-lived excited states of atoms and molecules in the gas, is additional evidence of the presence in the plasma of effects other than those thermally induced. Such a long-lasting plasma effect on the shock structure was also observed in the experiments by Bivolaru and Kuo21,22. The research in plasma mitigation of shock waves has two primary goals: 1). Improves the effective aerodynamic shape of an aircraft, but without the cooling requirements of a physical spike. 2). Results in reducing the shock noise and possibly a net energy savings.

3.

Plasma Spikes for the Mitigation of Shock Waves

Shock wave is formed by coherent aggregation of flow perturbations from an object. In the steady state, a sharp shock front signified by a step pressure jump is formed to separate the flow into regions of distinct entropies. The shock wave angle β depends on the Mach number M and the flow deflection angle θ through a θ-β-M relation25. Since the shock front is the far reachable edge of the flow perturbations deflected forward from an object, flow is unperturbed before reaching the shock front. To modify the shock structure and to move the shock wave upstream, the flow perturbations have to move upstream beyond the original shock front. One obvious way is to start the flow perturbation in front of the location of the original one, for instance, by introducing a longer physical spike. A plasma spike serves the same purpose, which encounters the flow in the region upstream of the location of the original shock front20. The induced flow perturbations from the plasma spike coalesce with the flow perturbations from the object into a new shock front, which replaces the original one located behind it. Charged particles accelerated by an electric field can deflect a neutral 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 the phenomenon of the transfer of charge 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 regime26. As shown in Fig. 3.34 on page 67 of Ref. 26, the charge transfer cross section between N2+ and N2 in the relevant energy regime is larger than 3 × 10-15 cm2. 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 don’t drift together, and thus the produced disturbances off the model 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 interact with the neutral flow through elastic collisions. The momentum transfer rate from electrons to neutrals is larger than that from ions to neutrals, i.e., νenmeve > νin 〈mi〉 vi, where 〈mi〉 = mimn/(mn + mi) ≅ mi/2. This can be easily verified by applying the relation νen/νin ≅ ve/vi, the drift speeds are usually larger than the corresponding thermal speeds in the pulsed discharges; thus νenmeve/ νin〈mi〉 vi ≅ 2meve2/mivi2 > 1, in most of arc discharges, where νen and νin are electron- and ion-neutral elastic collision frequencies. In airflow, the ion-neutral charge transfer collision frequency νc ~ 3νin, and 4

ve ~ eE/meνen and vi ~ eE/mi(νin + νc). Therefore, we can see that νenmeve ~ (νin + νc)mivi = (1 + νc/νin) νin mivi ~ 4 νin mivi. Considering flow deflection by a localized plasma as the shock wave modification mechanism; the flux distributions of the incoming airflow nnV0^ z (on the left) and the plasma electron flow neve (on the right), which represents a plasma spike, before collisions is presented in Fig. 1, in which the flux distribution in the middle of the figure represents the deflected flow after colliding with the plasma spike. This simple cartoon27 is used to demonstrate the deflection of airflow by a plasma spike.

Figure 1. Flux distributions of the incoming flow (left), the deflected flow (middle), and the electron flow (right) representing a plasma spike. 4.

Experimental Setup

Experiments were conducted in the test section of a 15 in × 15 in supersonic blowdown wind tunnel shown in Fig. 2. The upstream flow had a flow speed v = 570 m/s, temperature T1 = 135 K, and a pressure p1 = 0.175 atm. A 600-cone model shown in Fig. 3 was installed in the test section as a shock wave generator.

Figure 2. A Mach-2.5 wind tunnel.

5

This wind tunnel model is designed to have the tip and the body of the model as the two electrodes for gaseous discharge, which are separated by a ceramic insulator providing a 5-mm gap. The breakdown voltage is provided by a power supply, which includes a power transformer with a turns ratio of 1:25 to step up the line voltage of 120 V (rms) from a wall outlet to 3 kV (rms), an 1 µF capacitor in series with the electrodes, and a diode parallel to the electrodes to further step up the peak voltage. In the circuit, a 4 kΩ resistor is also connected in series with the diode preventing the charging current to exceed the specification of the diode. The circuit diagram is presented in Fig. 4.

Figure 3.

A photo of the cone model.

Figure 4. also shown.

A schematic of the circuit for the electric discharge; a voltage divider is

6

During one of the two half cycles when the diode is forward biased, the capacitor is charged and the voltage across the electrodes is low. During the other half cycle, the diode has a reverse bias. The charged capacitor increases the voltage across the electrodes and breakdown occurs. Since the discharge occurs only during one of the two half cycles, one can designate the tip electrode as a cathode or an anode by simply adjusting the orientation of the diode in the connection. However, plasma effect on the shock wave structure was observed only when the tip of the model was designated as the cathode. The discharge normally initiates in the region near the tip electrode where the applied electric field concentrates due to the cylindrical geometry. When the tip electrode is positive, it collects electrons produced by the discharge and increased electron transit-time loss shortens the arc path of the discharge. The discharge becomes asymmetric around the tip and generated plasma is distributed near the surface of the model19. The voltage across the electrodes has an asymmetric wave- form with a peak of about 4.5 kV, exceeding the 4 kV required for avalanche breakdown. The electric field intensity near the tip exceeds 10 kV/cm before breakdown occurs. It reduces to less than 1 kV/cm 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 a 60 Hz torch plasma, which was measured28 to have peak electron density exceeding 1013 electrons/cm3. During the run, the background pressure drops, thus the plasma density is expected to increase slightly. In Ref. 19, assuming 100% energy transfer for heating, an upper bound of the spatially averaged temperature rise was evaluated. The peak and average temperature enhancements were estimated to be ∆Tpeak ≅ 26 K (for Pin|peak = 1.2 kW) and ∆Tave ≅ 2.2 K (for Pin|average = 100 W). Therefore, the peak temperature perturbation is less than 10% of the unperturbed temperature 285 K in the region behind the shock, and less than 20 % of the ambient free stream gas temperature of 135 K, and the average temperature perturbation is less than 1% and 2 %, respectively. Even with the optimal estimation of 10 % peak temperature increase, it can only increase sound speed by 5% and decrease the Mach number by 5 %. Thus the temperature enhancement caused by the electric discharge was too low to introduce any significant thermal effects. 5.

Experimental Results

The produced spray-like plasma acted as a spatially distributed spike, which could deflect the incoming flow as that demonstrated by the cartoon shown in Fig. 1. A video camera was used to tape the shadow video graph of the flow. The shadowgraph technique is briefly described as follows. A uniform collimated light beam is introduced to illuminate the flow. The second derivative of the flow density deflects the light rays to a direction perpendicular to the light beam, which results in light intensity variation on a projection screen showing the shadow image of the flow field. Thus the location of a stationary shock front in the flow, where the second derivative of the density distribution is very large, is revealed in the shadowgraph as a dark curve because the light transmitted through that region is reduced to a minimum. The modification effect depends on the density and volume of the plasma spike produced by the discharge, which varies with time; in other words, the plasma spike increases its size and intensity from near zero to the maximum and then to decay to near zero. This time varying spike is expected to cause the shock front position to also vary in time. The video camera recorded

7

shadowgraph images at a rate of 30 frames per second and the pitch time was 50% of each frame time, which is slightly less than twice of each discharge period. Thus the temporal variation of the shock wave structure during a single discharge period cannot be recorded directly. However, continuous shadowgraphy of the flow can reconstruct the desired information regarding the transient behavior of the flow field. This is demonstrated in Fig. 5, which includes a sequence of six 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 presented in this section the flow is from left to right. The growth and decay of the plasma spike are manifested by the variation of the background brightness in the shadowgraphs. First shadowgraph shown in Fig. 5a 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 Figs. 5b and c corresponds to the case that the discharge is increasing before reaching the peak. The baseline shock front is first split into two (Fig. 5b), with a new one located upstream; the entire baseline shock front is then moved to the new one location as the plasma spike is intensified (Fig. 5c). As the plasma spike is further intensified to reach the peak, its modification effect on the shock structure also reaches the maximum. The shock front in Fig. 5d is very diffusive; it spreads from the one shown in Fig. 5c to further upstream region. The following two shadowgraphs in Figs. 5e and 5f are showing the recovery of shock wave from the plasma perturbation as the discharge decays from its peak to near zero. As the shock front moves upstream, its shock angle also increases. These observations on unsteady shock motion containing the above-mentioned features of the flow field (Figs. 5a to 5f) were typical of all the experiments performed and seen in the recorded videotape.

Figure 5. An assembled time sequence of six shadowgraphs (a)-(f) 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.

8

The pronounced influence of plasma on the shock structure is demonstrated by the result showing in Fig. 5d. A comparison of Figs. 5d and 5a clearly observes a transformation of the shock from a well-defined attached shock into a highly curved shock structure having a larger shock angle. The diffused form of the shock front could be an indication of shock wave being weakened by this plasma spike. Due to the spatial distribution and the non-uniformity of the generated plasma, the flow disturbances off the model become less coherent. Some of which eventually coalesce into shocks and others of which spread out in the form of expansion waves. Consequently, the shock front appears diffusively. The flow could also be perturbed to an asymmetric form. Thus the integration effect inherent in the shadowgraph technique when visualizing a three-dimensional flow field could also lead to the observed diffused form of the shock front. 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. This is an encouraging result, evidencing the effectiveness of this plasma scheme in reducing wave drag at supersonic speeds. A second video camera was used to record the spatial distribution and temporal evolution of the plasma through its airglow image. The results extracted from the videotapes of the shadow image of the flow and the airglow images of the plasma provide crucial information on the correlation between plasma distribution and modification of shock structure. It was shown that significant plasma effect on shock wave appeared under two conditions29: 1) plasma is generated in the region upstream of the baseline shock front and 2) plasma has a symmetrical spatial distribution with respect to the axis of the model.

(a)

(b)

(c)

(d)

Figure 6. (a) Plasma produced in front of the model with a symmetric distribution around the tip of the model, and (b) the shadowgraph of the flow field showing strong plasma effect on shock wave; (c) plasma has an asymmetric distribution around the tip of the model, and (d) the shadowgraph of the flow field showing negligible plasma effect on shock wave. 9

This conclusion is exemplified by comparing the two sets of the video graphs presented in Fig. 6. In the first set, Fig. 6a shows a spray-like plasma distributed symmetrically around the tip of the model. The flow direction points to the left. Significant plasma effect on shock wave is clearly shown by the corresponding shadowgraph of the flow field presented in Fig. 6b. In the second set, the plasma plume shown in Fig. 6c has an asymmetrical distribution very close to the surface of the model. In this case, no noticeable plasma effect on shock wave could be seen in Fig. 6d, the corresponding shadowgraph of the flow field. It is noted that the discharge in this case was, in fact, stronger than the previous case as indicated by the more intense background light of the shadowgraph. 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.7. Fig. 7a 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 plasma-producing model. Fig. 7b 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 Figs. 7a and 7b 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

Figure 7. A sequence of shadowgraphs taken during a wind tunnel run at Mach 2.5 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 same run, and (d) at the time when the discharge is around the peak and shock wave is eliminated. 10

also interesting to note that the shock in front of the 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. 7c 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 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. Closer examination of Fig. 7c 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 the 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. As a final example, Fig. 7d demonstrates the effectiveness of the plasma in eliminating the shock near the model, an ultimately desirable result that has significant consequences in minimizing wave drag and shock noise at supersonic speeds. In summary, the experimental results represented by the shadowgraphs (Figs. 7b-d) of the flowfield show that the spray-like plasma has strong effect on the structure of the shock wave. It causes the shock front to move upstream toward the plasma front and to become more and more dispersed in the process (Figs. 7b and c). A shock-free state (Fig. 7d) is observed as the discharge is intensified. The deflection of the incoming flow by a symmetrically distributed plasma spike in front of the shock is considered to be a likely process modifying the shock wave structure.

6.

Theory and Results

Two necessary conditions for achieving non-thermal plasma effect on shock waves are deduced from the experimental results18-21. The first one requires that plasma be generated in the region upstream of the baseline shock front, and the second one requires plasma to have a symmetrical spatial distribution with respect to the axis of the model29. Therefore, a symmetrically distributed plasma in front of the base-line shock, acting as a spike to deflect the flow27,30,31, will be incorporated into the formulation of the theory for understanding the experimental observations. Plasma electrons are kept in place by the local electric field in front of the tip of the cone. The transverse momentum perturbation of the neutral flow is distributed symmetrically in opposite directions. Thus this transverse perturbation can be very large even in the situation that plasma has very low ionization percentage and the electron mass is much smaller than those of neutral particles in the flow. This is because the net change of the total momentum in the transverse direction is zero and in the flow direction is small so that the total momentum of the system is conserved. 6.1. Supersonic flow over a cone We consider an azimuthally symmetric situation, where the incoming flow from the left propagates at an 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. 8.

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Figure 8.

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

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. 8 is adapted from Fig. 10.4 in the book by Anderson25, which is in the special case of θ′ = 0. The conic shock has an angle β to be determined. In region 2 immediately behind a the shock front, the flow has a deflection angle δ with Mach number M2 and velocity V2 = VR2 ^ a θ, where ^ a R and ^ a θ are unit vectors in spherical coordinate system, the origin is at the R − Vθ2 ^ tip of the cone, and the z axis is on 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 Anderson25. 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 = M1sin(β − θ′) and Mn2 = M2sin(β − δ). Letting β′ = (β − θ′) and δ′ = (δ − θ′), these two relations become Mn1 = M1sinβ′ and Mn2 = M2sin(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 25,27) is derived to be tan δ′ = 2 cotβ′{(M12sin2β′ − 1)/[M12(γ + cos2β′) + 2]}

(1)

where γ is usually chosen to be 1.4. The normalized Taylor-Maccoll equation for conical flows (Eq. 10.15 of Anderson25) is expressed as 0.2 [1 − G2 − G′2][2G + G′ cotθ + G″] − G′2[G + G″] = 0 where G = VR2/V2max, G′ = dG/dθ, G′′ = d2G/dθ2, and γ = 1.4 is assumed. The boundary conditions of (2) are given by

12

(2)

G(β) = f(M2) cos(β− δ) and G′(β) = −f(M2) sin(β − δ)

(3)

−1/2

+ 1] , M2 = Mn2/sin(β − δ), Mn2 = {[(M1sinβ′)2 + where f(M2) = V2/V2max = 5]/[7(M1sinβ′)2 − 1]}1/2, and δ is determined by (1). Eq. (2) will be solved based on the numerical procedure outlined in section 10.4 of Chapter 10 in the Anderson book25. That is, a proper β = βc is found by iteration in solving (2) such that the normal component of the flow velocity on the cone surface Vθ2(θc) = VR2′(θc) = G′(θc)V2max 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. [(5/M22)

6.2. Plasma spike and flow deflection This plasma spike is introduced at a location in front of the model by an on-board electrical discharge, which is triggered by a negative voltage applied between the grounded body of the cone and the tip of the cone located at z = 0, which is insulated from the body. The electric field at a point of distance R = (z2 + r2)1/2 away from the tip is given by E ≅ -(A0/R2)( zz ^ + rr ^ ), where A0 is proportional to the applied voltage and the tip is located at the origin. The electric field distributed in front of the tip accelerates plasma electrons in the upstream region to keep their energy much higher than that of the neutral particles. Some of them will gain enough energy for ionization, and the rest will deflect the incoming flow via momentum transfer collisions. Electrons cause more perturbation of the velocity distribution of the flow than ions. This is because the flow collides with electrons much more frequently than with ions. Moreover, the local field, in the region upstream of the tip of the model, accelerates ions in the direction of the flow, rather than opposite to the flow. We now consider that a uniform airflow from left to right with a velocity Vn0 = Vn0^ z encounters this plasma spike at z = - L. The plasma spike is generated by the discharge (L is the effective length of the plasma spike), before the airflow notices the presence of a cone on its way. Plasma is generated by the gaseous discharge in the imposed electric field E = ^ z Ez + ^ r Er 2 2 ≅ -[A0/(z + r )](zz ^ + rr ^ ). Electrons in the discharge interact with the airflow through elastic electron-neutral collisions. On the other hand, the ion-neutral interaction is more complicated. It involves both elastic and charge transfer inelastic collisions. The two-dimensional distribution of the electron flux neve(r) at z = - L in the azimuthally symmetric case is derived in a similar way as that in the previous work27,30,31. First, the electron density distribution of the spike is determined through the spatial distribution of the ionization frequency νi ~ ε5.3νa, where ε = E/Ecr, Ecr is the breakdown threshold field, and νa is the attachment rate. Thus ne(ξ) = ne0exp[(νi νa)t0] = n0exp{-η[1 − (1 + ξ2)−2.65] }, where ξ = r/L, t0 is the transient period for the plasma density to build up, n0 = ne0exp[(νi0 - νa)t0], νi0 = νi(ξ=0), η = νi0t0, and η = 0.85 will be assumed. The two electric field components in the interaction region at z = - L are represented approximately by Ez = E0/(1 + ξ2) and Er = -E0ξ/(1 + ξ2), where E0 = A0/L. The momentum equations for the three fluids: electrons, positive ions and neutral molecules, in this weakly ionized plasma in the presence of the imposed electric field are: med(neve)/dt = −nemeνen(ve – Vn) + nemeνei(vi - ve) − eneE

(4)

mid(nivi)/dt = −nimiνin(vi – Vn) −nimiνc(vi – Vn0) − nemeνei(vi - ve) + eniE

(5)

mnd(nnVn)/dt = nemeνen(ve – Vn) + nimiνin(vi – Vn) + nimiνc(vi – Vn0)

(6)

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where νc (> 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 Vn0, 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 also will not contribute to the shock wave formation. Therefore, each of the ion and neutral fluids can be decomposed into two components, i.e., niVi = ni1Vi1 + ni2Vi2 and nnVn = nn1Vn1 + nn2Vn2. The one with subscript 1 is related to shock formation, and the other, with subscript 2, is not related to shock formation. dni2/dt = dnn2/dt = νc ni1, and nn2 θcmax, there exists no Taylor-Maccoll solution, and the shock becomes detached25. θcmax decreases with the decrease of M1 and the equivalent θc of the cone increases with the increase of θ′. As shown in Fig. 9, M1 decreases and θ′ increases as ζ decreases (i.e., plasma density increases). ζ = 0.77 is the lowest value for which a TaylorMaccoll solution can be obtained. In other words, the numerical analysis indicates that the shock becomes detached when n0 > 7.5 × 1013 cm−3 (i.e., ζ < 0.77). Its standoff distance from the model is expected to increase with a further increase of n0. Detachment18 of the shock with increasing standoff distance from the model, as the discharge was intensified, is shown in Fig. 11, which is a superimposed shadowgraph (from the three shadowgraphs presented in Figs. 7a to 7c), showing the (attached) baseline shock front and two detached shock fronts modified by the plasma. The detached shock fronts appear more dispersive than the attached ones (which are also modified by the plasma) presented in Figs. 5d and 6b. In fact, it was observed19 in the same sequence of shadowgraphs presented in Fig. 7 that shock was eliminated near the peak of the discharge (Fig. 7d). This theory facilitates the understanding of the experimental observations, which cannot be reasonably explained by a simple heating effect.

Figure 11. A superimposed shadowgraph (from the three shadowgraphs presented in Figs. 7a to 7c) showing the (attached) baseline shock front and two detached shock fronts modified by the plasma with increasing discharge intensity. Acknowledgement This work was supported in part by Air Force Office of Scientific Research (AFOSR) Grant AFOSR-FA9550-04-1-0352.

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