_{1}

^{*}

When the spacecraft flies much faster than the sound speed (~1200 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 waves have been a detriment for the development of supersonic aircrafts, which have to overcome high wave drag and surface heating from additional friction. Shock wave also produces sonic booms. The noise issue raises environmental concerns, which have precluded routine supersonic flight over land. Therefore, mitigation of shock wave is essential to advance the development of supersonic aircrafts. A plasma mitigation technique is studied. A theory is presented to show that shock wave structure can be modified via flow deflection. Symmetrical deflection evades the need of exchanging the transverse momentum between the flow and the deflector. The analysis shows that the plasma generated in front of the model can effectively deflect the incoming flow. A non-thermal air plasma, generated by on-board 60 Hz periodic electric arc discharge in front of a wind tunnel model, was applied as a plasma deflector for shock wave mitigation technique. The experiment was conducted in a Mach 2.5 wind tunnel. The results show that the air plasma was generated symmetrically in front of the wind tunnel model. With increasing discharge intensity, the plasma deflector transforms the shock from a welldefined attached shock into a highly curved shock structure with increasing standoff distance from the model; this curved shock has increased shock angle and also appears in increasingly diffused form. In the decay of the discharge intensity, the shock front is first transformed back to a well-defined curve shock, which moves downstream to become a perturbed oblique shock; the baseline shock front then reappears as the discharge is reduced to low level again. The experimental observations confirm the theory. The steady of the incoming flow during the discharge cycle is manifested by the repeat of the baseline shock front.

Shock wave appears in the form of a steep pressure gradient. It introduces a discontinuity in the flow properties at the shock front location. The background pressure behind the shock front increases considerably, leading to significant enhancement of the flow drag and friction on the spacecraft. Thus, the design for high-speed aircraft tends to choose slender shapes to reduce the drag and cooling requirements. This is an engineering tradeoff between volumetric and fuel consumption efficiencies and this tradeoff significantly increases the operating cost of commercial supersonic aircraft. Shock wave also produces sonic booms. The noise issue raises environmental concerns, which have precluded routine supersonic flight over land.

Theoretical and experimental efforts have been devoted to the understanding of shock waves in supersonic flows [

Thermal energy deposition in front of a flying body to perturb the incoming flow and shock wave formation has been studied [

Non-thermal modification effects of plasmas on the shock wave structures have been evidenced in a number of shock-tube experiments. The study of plasma mitigation of shock waves is further inspired by the observation of a wind tunnel experiment conducted by Gordeev et al. [^{ }

A long-lasting plasma effect on the shock structure (i.e., it takes much longer than the discharge period, after the discharge ceases, to recover to the baseline state) was also observed in the experiments by Baryshnikov et al. [

Theory shows that the shock wave angle and the shock structure depend on the cone angle of the wind tunnel model, and on the Mach number and the deflection angle of the incoming flow [

In Section 2, the Taylor-Maccoll’s theory for a normally incident supersonic flow over a cone is generalized to the case of obliquely incident flow. A plasma deflector generated by an electric discharge is modeled and the flow deflection by this deflector is formulated in Section 3. Numerical illustration of the plasma deflection effect is presented in Section 4. Wind tunnel experimental results to demonstrate shock mitigation by a plasma deflector, generated by 60 Hz periodic arc discharge, are presented in Section 5. In Section 6, a summary of the work is given. Conclusion remarks are drawn in Section 7.

Starting with a simple situation that the incoming supersonic flow from the left propagates along the axis of a 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 _{1} in region 1 is along the z (cone’s) axis, i.e., V 1 = V 1 a ^ z = V 1 ( a ^ R cos β − a ^ θ sin β ) , where a ^ R and a ^ θ are unit vectors in the radial and poloidal directions of the spherical coordinate system, the origin is at the tip of the cone. The flow has a Mach number M_{1}. The conic shock wave angle β is to be determined. In region 2 immediately behind the shock front, the flow has a deflection angle δ with Mach number M_{2} and velocity V 2 = V 1 [ a ^ R cos ( β − δ ) − a ^ θ sin ( β − δ ) ] . Across an oblique shock wave, the continuity of the flow, i.e., ρ 1 V 1 sin β = ρ 2 V 2 sin ( β − δ ) where ρ is the mass density of the flow, together with the preservation of the tangential component (i.e. a ^ R component) of the flow velocity, i.e., V 1 cos β = V 2 cos ( β − δ ) , relate the shock wave angle β and the deflection angle δ through a δ-β-M relation [

tan δ = 2 cot β { ( M 1 2 sin 2 β − 1 ) / [ M 1 2 ( γ + cos 2 β ) + 2 ] } (1)

where γ = 1.4 is usually adopted.

Taylor-Maccoll’s theory is applied to analyze a deflected supersonic flow over a cone. The incoming flow from the left is now to propagate at a constant angle θ ′ with respect to the axis of the cone as sketched in

δ ′ = ( δ − θ ′ ) , these two relations become M n 1 = M 1 sin β ′ and M n 2 = M 2 sin ( β ′ − δ ′ ) , which are expressions similar to those in the θ ′ = 0 case. In essence, this is to rotate the z axis counterclockwise by an angle θ ′ . Therefore, in the case of θ ′ ≠ 0 , the δ-β-M relation is extended to the δ'-β'-M relation [

tan δ ′ = 2 cot β ′ { ( M 1 2 sin 2 β ′ − 1 ) / [ M 1 2 ( γ + cos 2 β ′ ) + 2 ] } (2)

The normalized Taylor-Maccoll equation for conical flows (Eq. 10.15 of Anderson [

0.2 [ 1 − G 2 − G ′ 2 ] [ 2 G + G ′ cot θ + G ″ ] − G ′ 2 [ G + G ″ ] = 0 (3)

where G = V R 2 / V 2 max , G ′ = d G / d θ , G ″ = d 2 G / d θ 2 , and γ = 1.4 is assumed.

The boundary conditions of (3), at the poloidal angle θ = β , are given by

G ( β ) = f ( M 2 ) cos ( β − δ ) = f ( M 2 ) cos ( β ′ − δ ′ ) G ′ ( β ) = − f ( M 2 ) sin ( β − δ ) = − f ( M 2 ) sin ( β ′ − δ ′ ) (4)

where f ( M 2 ) = V 2 / V 2 max = [ ( 5 / M 2 2 ) + 1 ] − 1 / 2 , M 2 = M n 2 / sin ( β ′ − δ ′ ) ,

M n 2 = { [ ( M 1 sin β ′ ) 2 + 5 ] / [ 7 ( M 1 sin β ′ ) 2 − 1 ] } 1 / 2 , and δ ′ is determined by (2).

Thus G ( θ ; β ) and G ′ ( θ ; β ) can be evaluated for a given M_{1} and β. If β represents the wave angle of the shock on a θ half-angle cone, an additional boundary condition that the normal component of the flow velocity on the cone surface is zero, i.e., G ′ ( θ ; β ) = 0 , has to be imposed.

Equation (3) can be solved for G ( θ c ; β c ) , where θ_{c} and β_{c} are the half-cone angle and the wave angle of the shock on the cone, via a direct or an inverse approach. The direct numerical approach is to continuously adjust the boundary conditions by varying β until G ′ ( θ c ; β c ) = 0 with a proper boundary angle β = β c . From this, the wave angle β_{c} of the shock on a θ_{c} half-angle cone is determined. However, it makes the calculation easy by employing an inverse approach; a given shock wave will be assumed and the particular cone that supports the given shock will be calculated.

That is setting β = β ′ c , a given shock wave angle β ′ c , in (4) to solve (3) for G ( θ ; β ′ c ) and G ′ ( θ ; β ′ c ) = 0 until G ′ ( θ = θ ′ c ; β ′ c ) = 0 . Thus a plot of β ′ c verse θ ′ c can be obtained; from which ( θ c , β c ) is determined. The effect of a localized plasma deflector on the shock wave is then inferred from changes in the deflection angle θ ′ and in the Mach number M_{1} of the flow, where θ ′ and M_{1} vary with r, the radial coordinate with respect to the z-axis.

At a fixed M_{1}, the deflection angle δ and the shock wave angle β, in the case of θ ′ = 0 , increase only with the half-cone angle θ_{c}. This is illustrated in

Thus, the shock will become detached and the oblique shock becomes a curved shock, as shown in _{c} as well as with the deflection angle θ ′ of the incoming flow. It suggests that the shock wave angle of a fixed cone can be increased by introducing a symmetric deflection on the incoming flow; and further increase of the deflection can even modify the attached oblique shock to a detached curve shock. The impact of the flow deflection on the shock wave modification is illustrated in

To deflect the incoming flow effectively, it favors that plasma is generated in the region upstream of the baseline shock front and has a symmetrical spatial distribution with respect to the axis of a cone [

This plasma deflector is introduced at a location in front of a (cone-shaped) wind tunnel 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 sharpness of the tip helps to enhance the electric field intensity in the region in front of the tip. Using the planar projection of the model as a two-dimensional model, the equipotential lines between the two electrodes, with the central electrode biased negatively, are evaluated numerically by using a Poisson solver. The result is presented in _{0} is proportional to the applied voltage and R = ( z 2 + r 2 ) 1 / 2 is the distance of the field point away from the tip. Because electrons collide the incoming flow much more frequently than ions and the local field, in the region upstream of the tip of the model, accelerates ions in the direction of the flow, electrons deflect the flow stronger than the ions. The deflection is most effective when plasma has a symmetric distribution around the tip, so that the net change

of the total momentum of the deflected flow in the transverse direction is zero; thus this transverse perturbation can be large even in the situation that the electron mass is much smaller than the masses of the neutral particles in the flow.

We now consider that a uniform airflow from left to right with a velocity V 10 = V 10 z ^ encounters this plasma deflector at z = − z 0 , as demonstrated in

The electron density distribution of the plasma deflector is first determined through the spatial distribution of the ionization frequency ν i ~ ε 5.3 ν a , where ε = E / E c r , E_{cr} is the air breakdown threshold field, and ν_{a} is the electron-neutral

attachment rate. Thus n e ( ξ ) = n 0 exp [ ( ν i − ν a ) t 0 ] = n e 0 exp { − η [ 1 − ( 1 + ξ 2 ) − 2.65 ] } , where ξ = r / z 0 , t_{0} is the transient period for the plasma density to build up, n e 0 = n e ( 0 ) = n 0 exp [ ( ν i 0 − ν a ) t 0 ] , ν i 0 = ν i ( ξ = 0 ) , η = ν i 0 t 0 , and η = 0.85 will be assumed. The two electric field components in the interaction region at z = − z 0 are represented approximately by E z = E 0 / ( 1 + ξ 2 ) and E r = − E 0 ξ / ( 1 + ξ 2 ) , where E 0 = A 0 / z 0 .

The momentum equations for the three fluids: electrons, positive ions and neutral molecules, in a weakly ionized plasma in the presence of an imposed electric field are:

m e d ( n e v e ) / d t = − n e m e ν e n ( v e − V 1 ) + n e m e ν e i ( v i − v e ) − e n e E (5)

m i d ( n i v i ) / d t = − n i m i ν i n ( v i − V 1 ) − n i m i ν c ( v i − V 10 ) − n e m e ν e i ( v i − v e ) + e n i E (6)

m n d ( n n V 1 ) / d t = n e m e ν e n ( v e − V 1 ) + n i m i ν i n ( v i − V 1 ) + n i m i ν c ( v i − V 10 ) (7)

where ν_{c} ( > ν_{in}) is the ion-neutral charge transfer collision frequency, which is dominated by charge transferring between the same type particles, e.g., the charge transfer cross-section between N 2 + and N_{2} in the relevant energy regime is larger than 3 ´ 10^{−19} m^{2}. After charge transfer, neutral particle converts to an ion which moves at the neutral’s original velocity V_{10}; and the ion converts to a neutral moving at ion’s original velocity v_{i}_{0}. 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; but this ion flow will be collected by the cathode to close the discharge current loop, and will not contribute to the shock wave formation. Therefore, each of the ion and neutral fluids can be decomposed into two components, i.e., n i v i = n i 1 v i ( 1 ) + n i 2 v i ( 2 ) and n n V 1 = n n 1 V 1 ( 1 ) + n n 2 V 1 ( 2 ) , where n_{i}_{1} and n_{i}_{2} are the ion densities generated by the electric discharge and by the charge exchange conversion, respectively; n_{n}_{1} and n_{n}_{2} are the neutral density of the incoming flow and the neutral density converted from the ions, respectively; thus d n i 2 / d t = d n n 2 / d t = ν c n i 1 ; and v i ( 1 ) ~ v i 0 , v i ( 2 ) ~ V 10 , V 1 ( 1 ) ~ V 10 , and V 1 ( 2 ) ~ v i 0 . In other words, the one with superscript 1 is related to shock formation, and the other, with superscript 2, is not related to shock formation. n n 2 ≪ n n 1 because the interaction region (plasma layer) is very narrow. To simplify the analysis, the inertial terms on the left hand side and the electron-ion collision terms (proportional to ν_{ei} and ν_{ie}) on the right hand side of (5) and (6) are neglected; the approximated (5) and (6)

yield n e m e ν e n ( v e − V 1 ( 1 ) ) ≅ − n e e E and

n i 1 m i ν i n ( v i ( 1 ) − V 1 ( 1 ) ) ≅ [ ν i n / ( ν i n + ν c ) ] n i 1 ( e E − m i ν c V 1 ( 1 ) ) , which reduce (7) to be

d V 1 ( 1 ) / d t = − [ ν c / ( ν i n + ν c ) ] ( n e / n n ) e E / m n − ( n i 1 / n n ) ν c [ V 10 + ν i n V 1 ( 1 ) / ( ν i n + ν c ) ] (8)

where n i 1 ≅ n e is assumed.

It is noted that in (8) 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 deflector. We now integrate (8) over a transit period t n = z 0 / V 10 , the time for the airflow to pass through the plasma deflector of length z_{0}. We obtain

V 1 ( 1 ) ( r , t n ) ≅ V 10 z ^ − V 10 exp { − η [ 1 − ( 1 + ξ 2 ) − 2.65 ] } [ α n ( 1 + ξ 2 ) − 1 ( z ^ − ξ r ^ ) + β n z ^ ] (9)

where α n = [ ν c / ( ν i n + ν c ) ] ( e E 0 t n / V 10 m n ) ( n e 0 / n n ) and β n = ( n e 0 / n n ) ( ν c t n ) [ 1 + ν i n / ( ν i n + ν c ) ] . Thus the deflected flow has spatially dependent deflection angle θ ′ and Mach number M_{1}, which are obtained from (9) to be θ ′ ( r ) = tan − 1 [ V 1 r ( r ) / V 1 z ( r ) ] and M 1 ( r ) = { [ V 1 r ( r ) 2 + V 1 z ( r ) 2 ] 1 / 2 / V 10 } M 10 , where M_{10} is the Mach number of the unperturbed flow.

Consider a case comparable with the experiment to be presented in the next section, the parameters of the incoming flow used in the numerical calculations are: M_{10} = 2.5, n_{n} ~ 10^{25} m^{−3} (i.e., P_{1} = 0.175 atm and T_{1} = 135 K), V_{10} = 570 m/s, and ν c ≅ 2 ν i n ≅ 2.16 × 10 9 sec − 1 .

The deflection angle θ ′ ( r ) and the Mach number M_{1}(r) of the deflected flow vary with the intensity of the discharge (depending on the applied electric field intensity and gauged by the maximum electron density n_{e}_{0}). Set E_{0} ~ 10^{6} V/m and t_{n} = 0.88 ´ 10^{-}^{5} s (i.e., z_{0} = 5 mm), the parameters a_{n} and β_{n} in (9) become α n = 0.334 × 10 − 20 n e 0 and β n = 0.253 × 10 − 20 n e 0 . Let n e 0 = 3 × 10 20 ζ m^{-}^{3}, where z is a variable parameter to weigh the discharge situation, so that a_{n} = z and β_{n} = 0.76z. The two functions θ ′ ( r ) and M_{1}(r) are plotted in

Using these results for each z (i.e., n_{0}) as the parameters at the shock front location, the corresponding oblique angle β c ( r ) = β ′ c + θ ′ of the shock front can

be determined by solving (3) iteratively to meet the condition that the normal component of the flow velocity on the cone surface G ′ ( θ c ; β c ) = 0 . Thus the position of the shock front can be determined by the trajectory equation

d z / d r = cot β c = cot ( β ′ c + θ ′ ) (10)

The result in the case of z = 0.2 (corresponding to a relatively intense discharge) is presented in _{0} = 42.6˚ (for θ_{c} = 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 at the left lower corner of the same figure for comparison. In this case, the peak electron density of the plasma deflector used in the numerical calculation is n_{e}_{0} = 6 ´ 10^{19} m^{−3}, which agrees with that produced by the on-board diffused arc discharge, shown in the insert at the left upper corner of the same figure.

For each z, a β_{c}(x) distribution is determined. In terms of the determined β_{c}(x) and δ for β ′ and δ ′ in (2), one can obtain an equivalent Mach number distribution M_{1eq}(x). This is the Mach number distribution for an undeflected flow (i.e., in the absence of the plasma deflector) 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 M_{1eq}(x) of the incoming flow in the tip region has a similar spatial distribution as the corresponding M_{1}(x) presented in _{10}.

Experiment was conducted in a Mach 2.5 wind tunnel [_{1} = 135 K, and a pressure P_{1} = 0.175 atm.

Implement a plasma torch module [_{b} = 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 a power supply sustaining 60 Hz periodic electric discharge. The nose of the model, shown in

plasma effect on the shock wave structure has been observed only when the tip of the model is designated as the cathode.

The output voltage of the power supply has 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 a level 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 maximum electron density of the discharge exceeds 10^{19} m^{−3}.

A video camera, as the corresponding one to the CCD camera used to record the shadowgraph images of the flowfield, is used to record the spatial distribution and temporal evolution of the plasma glow with the frame rate of 30 frames per second and exposure time of 1/60 s (which is slightly less than six times of each discharge period). The video graph recorded in each frame is an integrated result over the exposure time, and thus the temporal variation of the plasma glow during a single discharge period cannot be recorded directly; but it can extract from the continuous video graph of the discharge. A sequence of 6 assembled video graphs showing the growth and decay of the plasma glow in the electric discharge, with a symmetrical spatial distribution with respect to the axis of a cone, are presented in

Shadowgraph method is used to optically diagnose the flowfield around the spike and nose of the cone. A black and white charge coupled device (CCD) camera, also with a frame rate of 30 frames per second and exposure time of 1/60 s, is used to record directly the shadowgraph images of the flow structure. 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). The images extracted from videos recording the shadowgraph images of the flow and recording the plasma glow can provide the correlation between the strength of the plasma deflector and the degree of the shock structure modification; and any consistent relationships appearing in the correlation are useful for determining the plasma conditions, in order to achieve significant plasma effect on the shock wave.

The spray-like plasma, generated by the electric discharge, acted as a spatially distributed plasma deflector, which was used to deflect the incoming flow. A video camera was used to record the shadowgraph images of the flow. The modification effect depended on the density, volume, spatial distribution, and location of the plasma deflector produced by the electric discharge, which varied in

time; thus the plasma deflector increased its size and intensity from a low level to the maximum and then decayed to a low level, as demonstrated in

flow to the growth and decay of the plasma deflector in one discharge cycle. In these shadowgraphs, the flow is from left to right.

The growth of the plasma deflector is manifested by the variation of the background brightness in the shadowgraphs. First, shadowgraph shown in

The pronounced influence of plasma on the shock structure is demonstrated by the result shown in

A theory based on the deflection of the incoming flow by a symmetrically distributed plasma deflector in front of the shock as the process to modifying the shock wave structure has been formulated. Flow deflection increases the equivalent cone angle, which in turn increases the attached oblique shock wave angle. Moreover, when this equivalent cone angle exceeds a critical angle, the shock becomes a detached curved shock.

Wind tunnel experiments were conducted to explore the non-thermal plasma effects, based on plasma deflection theory, on the shock wave structures. A cone-shaped model was used as the shock generator and was facilitated with electrodes for on-board 60 Hz periodic discharge to generate plasma in front of the model. The tip of the central electrode in the model was shaped to match the cone angle and to enhance the electric field intensity in the region in front of the tip. The central electrode was set as the cathode. This arrangement together with the favorable electric field distribution made electron flow 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, set as the anode. As shown in

As shown in the sequence of 6 shadowgraphs presented in

The theoretical deflection model is analyzed numerically. As demonstrated in

The wave drag and noise of the shock on the cone depends on the strength of the shock, which in turn depends on the Mach number of the flow as well as the shock wave structure. It is found that the effective Mach number M_{1eq}(x) of the deflected flow in the tip region is smaller than the unperturbed one (M_{10}). A decrease in the effective Mach number of the incoming flow in the tip region verifies that the plasma deflector can indeed reduce the wave drag of the shock on a supersonic aircraft. Moreover, as the modified shock structure moves upstream away from a supersonic aircraft and becomes diffusive, it also results to the reduction of the wave drag on the supersonic aircraft and of the shock noise generated in the flow.

The author is grateful to Dr. D. Bivolaru for collaborative work. This work was supported in part by the Air Force Office of Scientific Research Grant AFOSR-FA9550-04-1-0352.

The author declares no conflicts of interest regarding the publication of this paper.

Kuo, S.P. (2018) Shock Wave Mitigation by Air Plasma Deflector. Advances in Aerospace Science and Technology, 3, 71-88. https://doi.org/10.4236/aast.2018.34006